Regulation Design in Insurance Markets: Bhaskar, McClellan & Sadler (2023)

What this is. The paper’s core propositions, the formal delegation model it builds on, and the latent-contract mechanism with its defining equations: enough to understand what was proved and how, without reading all 35 pages. To replicate or extend, read the full source at the original.

TL;DR

A regulator seeks to restrict the menus of insurance contracts a firm may offer. The firm privately observes a signal about each consumer’s type, giving it an informational advantage the regulator lacks. The paper’s main result (Theorem 1) shows that despite this asymmetry, the regulator can implement any socially optimal allocation by augmenting each intended menu with at most two “latent contracts”: high-coverage expensive options and low-coverage cheap options that are never chosen in equilibrium but deter the firm from offering menus designed for different signals. Under an order condition that higher signals correspond to higher consumer coverage need, these latent contracts can be constructed from the model’s primitives. A separate result shows that when the regulator maximizes consumer welfare, more firm information weakly improves welfare under optimal regulation: the regulator can turn the firm’s data advantage into a tool rather than a threat.

Core results

Locators reference the published version. No numerical magnitudes are reported; all results are theoretical propositions.

#	Result	Locator	Formal claim
R1	Main theorem: any incentive-compatible, participation-feasible allocation can be implemented with at most two latent contracts per menu	Theorem 1, p. 2560	There exist contracts {c̄^s, c_s}{s∈S} such that policy R = {M^s{a*} ∪ {c̄^s, c_s}}_{s∈S} implements a* in perfect Bayesian equilibrium
R2	Noncontractible loss model: optimal allocation gives all agents in category x the same contract, equalizing marginal utility across all types and events	Prop. 1(i), pp. 2561-2562	c_x = (p*, t_x*) where t_x*(ω) equates the marginal utility of transfers u_z across all (ω, x) pairs
R3	Under supermodularity of utility in transfer and category, transfers are strictly increasing in category and one downward latent contract per menu suffices	Prop. 1(ii), p. 2562	t_x*(ω) strictly increasing in x for each ω; policy R = {M_x}_{x∈X} with M_x = {c_x, c̄_x} implements a*
R4	Consumer welfare is weakly higher under optimal regulation when the firm has more information	Prop. 2, p. 2563	If S’ is more informative than S, an optimal policy under S’ attains W ≥ the optimal W under S
R5	Local improvements: any nearby allocation can be implemented with latent contracts proportionally close to existing contracts	Prop. 3, pp. 2564-2565	There exists K ≥ 1 such that for any Δ > 0 and any allocation â with d(â, a) < Δ satisfying agent incentives and firm participation, there is a policy R̂ with D(R̂, R) ≤ KΔ implementing â
R6	Price cap plus latent contracts: a data-driven price cap on the highest-risk menu, augmented with latent contracts on all other menus, robustly improves on the laissez-faire outcome using only observable market data	Prop. 5, pp. 2568-2569	For menu M_n (highest signal), price-capped menu M^C_n combined with latent contracts c̄^i on menus M_1,…,M_{n-1} leads the firm to offer M^C_n after signal s_n; highest-risk types purchase full coverage at a lower price p^C < p^{m_n}_n

Overall (paper’s conclusion). Regulators can leverage a firm’s informational advantage against itself. By requiring firms to include off-path latent contracts in each menu, the regulator implements her preferred allocation without observing the firm’s signal, and the mechanism requires each consumer to face at most three contracts. This insight extends to price caps and other standard regulatory tools when combined with targeted latent contracts inferred from market data.

Theory / model

The framework builds on the monopolistic insurance screening model of Stiglitz (1977), in which a firm offers menus to separate risk types, and on the delegation theory of Holmstrom (1984), who studies a principal restricting the action set of a better-informed agent. Baron and Myerson (1982) study optimal price regulation of a monopolist with privately known costs; the key difference here is that the firm contracts with individual consumers and screens them using its data advantage. Chade and Schlee (2012) characterize optimal insurance allocation under adverse selection; the noncontractible loss model below embeds their canonical formulation as a special case. Galperti (2015) shows commitment devices can screen time-inconsistent agents; latent contracts serve an analogous off-path incentive role here. Brunnermeier, Lamba, and Segura-Rodriguez (2020) study how a firm better informed than consumers designs profit-maximizing menus; this paper adds a regulator who turns that asymmetry into a welfare tool.

The model is a three-stage game (regulator, firm, consumers) with heterogeneous agents and a privately informed intermediary (pp. 2554-2556).

Types and preferences. Each agent has a two-dimensional type $\tau = (x, \theta) \in T = X \times \Theta$, where $X \subset \mathbb{N}$ is the finite set of categories and $\Theta \subset \Delta(\Omega)$ is the set of risk types. Categories capture utility differences in the event of a loss (treatment cost, severity of need); risk types determine the probability distribution over verifiable loss events $\Omega$. Types are distributed according to $\mu \in \Delta(X \times \Theta)$. A contract $c = (p, t)$ specifies a premium $p \in \mathbb{R}_+$ and a transfer function $t : \Omega \to \mathbb{R}_+$. Agent $\tau$ buying contract $c$ obtains expected utility (p. 2554):

$$U(\tau, c) := \sum_{\omega \in \Omega} \theta(\omega)\, u(t(\omega) - p,\, \omega,\, x) \tag{1}$$

where $u : \mathbb{R} \times \Omega \times X \to \mathbb{R}$ is strictly increasing and strictly concave in its first argument, with the Inada condition $\lim_{z \to \underline{z}_{\omega,x}} u_z = \infty$. The firm’s expected profit from selling $c$ to type $\tau$ is:

$$\Pi(\tau, c) := p - \sum_{\omega \in \Omega} \theta(\omega)\, t(\omega) \tag{2}$$

Information structure. The firm privately observes a signal $s \in S \subset \mathbb{N}$ about each agent’s type; the regulator observes neither the signal nor the type, only the aggregate joint distribution of types and signals. Let $\mu(\cdot \mid s)$ denote the conditional type distribution given signal $s$ and $\text{supp}(s)$ the support of $\mu(\cdot \mid s)$.

Regulatory policy and timing. A regulatory policy $\mathcal{R}$ is a set of menus (each menu is a set of contracts). Timing: (i) the regulator chooses $\mathcal{R}$; (ii) the firm decides whether to enter, then offers each agent a menu from $\mathcal{R}$ based on its signal; (iii) agents choose a contract from the offered menu; (iv) loss events realize and transfers are paid.

Aggregate welfare. An allocation $\mathbf{a} = \{c^s_\tau\}$ specifies, for each type and signal, a contract. Total consumer welfare and firm expected profit from allocation $\mathbf{a}$ are (p. 2556):

$$W(\mathbf{a}) := \sum_{\tau \in T} \mu(\tau) \sum_{s:\, \tau \in \text{supp}(s)} \Pr(s \mid \tau)\, U(\tau, c^s_\tau) \tag{3}$$ $$\pi(\mathbf{a}) := -k + \sum_{\tau \in T} \mu(\tau) \sum_{s:\, \tau \in \text{supp}(s)} \Pr(s \mid \tau)\, \Pi(\tau, c^s_\tau) \tag{4}$$

The regulator maximizes a social welfare function $F(\mathbf{a})$ satisfying $F(\mathbf{a}) = -\infty$ whenever $W(\mathbf{a}) = -\infty$; a canonical special case is $F(\mathbf{a}) = \beta W(\mathbf{a}) + (1-\beta)\pi(\mathbf{a})$ for $\beta \in (0,1]$.

Order condition (Assumption 1, p. 2558). The main result requires that higher signals correspond to higher coverage need. There exists a loss event $\omega_1 \neq \omega_0$ such that $u(z, \omega, x)$ is supermodular in $z$ and $x$, and for every signal $s$, a maximal type $\bar{\tau}^s = (\bar{x}^s, \bar{\theta}^s)$ and a minimal type $\underline{\tau}^s = (\underline{x}^s, \underline{\theta}^s)$ exist in $\text{supp}(s)$ such that:

• the maximal type has a weakly higher category and higher likelihood ratio $\bar{\theta}^s(\omega_1)/\bar{\theta}^s(\omega_0) \geq \theta(\omega_1)/\theta(\omega_0)$ for all types in the support of $s$; and
• for $s' > s$, the maximal category $\bar{x}^{s'} \geq \bar{x}^s$ and the maximal likelihood ratio in signal $s'$ is strictly greater than in signal $s$.

This means higher signals are supported on types with weakly higher category and strictly higher maximal risk, capturing that individuals with poorer health or greater need are more likely to generate higher firm signals.

Relaxed problem. The regulator first solves a relaxed problem (RP) that ignores firm incentive constraints, treating the firm as if the regulator could directly observe the signal (pp. 2559-2560):

$$\max_{\mathbf{a}}\; F(\mathbf{a}) \quad \text{s.t.} \quad \pi(\mathbf{a}) \geq 0 \text{ and } U(\tau, c^s_\tau) \geq U(\tau, c)\; \forall\, \tau \in \text{supp}(s),\, c \in M^s_\mathbf{a},\, s \in S \tag{RP}$$

The firm’s incentive constraints to truthfully offer the menu intended for each signal are then the binding constraint for implementation in the actual problem.

Noncontractible loss model (special case). For the main application, each verifiable event $\omega$ contains a set of noncontractible states $\hat{\omega}$; agents in category $x$ face the distribution $\nu_x(\hat{\omega} \mid \omega)$ over them, suffering loss $\tilde{\ell}(\hat{\omega})$. Utility in event $\omega$ is (p. 2557):

$$u(t(\omega) - p,\, \omega,\, x) := \int_{\hat{\omega} \in \omega} v\!\left(e - p + t(\omega) - \tilde{\ell}(\hat{\omega})\right) d\nu_x(\hat{\omega} \mid \omega) \tag{5}$$

where $e$ is initial wealth and $v$ is strictly concave. In the canonical Stiglitz (1977) model this collapses to two states (no loss, loss $\ell$) with a single category $x$.

Method

The proof of Theorem 1 constructs two families of latent contracts for each signal $s$, using the order condition to ensure they deter deviations without distorting on-path allocations.

Latent contract properties (p. 2560). For each signal $s$, the two contracts $(\bar{c}^s, \underline{c}^s)$ added to menu $M^s_{a^*}$ satisfy:

1. Every type $\tau \in \text{supp}(s)$ weakly prefers the allocated contract $c^s_{\tau, a^*}$ to both latent contracts.
2. For any higher signal $s' > s$, all types $\tau \in \text{supp}(s')$ with $\theta(\omega_1)/\theta(\omega_0) > \bar{\theta}^s(\omega_1)/\bar{\theta}^s(\omega_0)$ and $x \geq \bar{x}^s$ prefer the downward latent contract $\bar{c}^s$ over any contract in $M^s_{a^*}$.
3. For any lower signal $s' < s$, all types $\tau \in \text{supp}(s')$ with $\theta(\omega_1)/\theta(\omega_0) < \underline{\theta}^s(\omega_1)/\underline{\theta}^s(\omega_0)$ and $x \leq \underline{x}^s$ prefer the upward latent contract $\underline{c}^s$ over any contract in $M^s_{a^*}$.

Construction of the downward latent contract (Lemma 1, p. 2574). Fix signal $s$ and let $(\bar{x}^s, \bar{\theta}^s)$ be the maximal type in its support. Let $(p', t')$ be a reference contract with $U(\tau, (p', t')) > -\infty$ for all $\tau$. The downward latent contract $(\bar{p}, \bar{t})$ is constructed so that the maximal type is indifferent between $(\bar{p}, \bar{t})$ and $(p', t')$. The indifference condition is (equation A1, p. 2574):

$$\frac{\bar{\theta}^s(\omega_1)}{\bar{\theta}^s(\omega_0)} \Big[ u\!\left(\hat{t} - \bar{p},\, \omega_1,\, \bar{x}^s\right) - u\!\left(t'(\omega_1) - p',\, \omega_1,\, \bar{x}^s\right)\Big] = u\!\left(t'(\omega_0) - p',\, \omega_0,\, \bar{x}^s\right) - u\!\left(-\bar{p},\, \omega_0,\, \bar{x}^s\right) \tag{A1}$$

For fixed $\hat{t} > t'(\omega_1)$, a unique $\bar{p} \in [p' - t'(\omega_0),\, -\underline{z}]$ solving (A1) exists because the right-hand side is finite and decreasing in $\bar{p}$, and utility is unbounded above. Raising $\hat{t}$ drives the firm’s expected profit from selling $(\bar{p}, \bar{t})$ toward $-\infty$, because $\lim_{\hat{t} \to \infty} \bar{p}'(\hat{t}) = 0$ (the marginal increase in $\bar{p}$ vanishes, so $\hat{t} - \bar{p}(\hat{t}) \to \infty$) while the transfer cost $\bar{\theta}^s(\omega_1) \hat{t} \to \infty$ (p. 2574). Choosing $\hat{t}$ large enough makes the downward latent contract arbitrarily unprofitable for the firm, so no signal $s' < s$ is worth offering after signal $s$. By supermodularity, any type in a higher signal’s support who could prefer $\bar{c}^s$ over the intended contract is exactly the kind of type the regulator wants to attract away from the wrong menu.

An analogous upward latent contract $(\underline{p}, \underline{t})$ is constructed with lower payouts in $\omega_1$ and a higher transfer in $\omega_0$, targeting the minimal type $\underline{\tau}^s$, to deter the firm from offering a lower-signal menu after signal $s$.

Equilibrium verification (proof of Theorem 1, pp. 2576-2577). With $M^s = M^s_{a^*} \cup \{(\bar{p}^s, \bar{t}^s), (\underline{p}^s, \underline{t}^s)\}$, consider the strategy profile: the firm offers $M^s$ after signal $s$; on-path agents choose their allocated contract; if the firm deviates to $M^{s'}$, all types who weakly prefer the corresponding latent contract choose it. For $\hat{t}$ sufficiently large, the latent contracts are chosen by enough types that the deviation is unprofitable regardless of what other types choose, since the expected loss from latent-contract purchases is unbounded. Hence the firm’s optimal response is to follow the regulator’s prescribed menu after each signal.

Data-driven price cap (Proposition 5, pp. 2568-2569). For the canonical one-category model the regulator can observe three objects from laissez-faire market data: the contracts sold in each menu $M_i = \{(p_i^1, t_i^1), \ldots, (p_i^{m_i}, t_i^{m_i})\}$; the fraction $\lambda(c_i^k)$ of agents choosing each contract; and the fraction $\rho(c_i^k)$ suffering a loss. From these she infers: the loss amount $\ell = t_n^{m_n}$ (the transfer paid by the full-coverage contract on the highest-signal menu); the highest-risk types $\bar{\theta}^{s_i} = \rho(c_i^{m_i})$; and their menu share $\lambda(c_n^{m_n})$. The price-capped menu $M_n^C$ charges $\bar{p}_i = \min\{p_i, p^C\}$ for each contract. Adding to each other menu a latent contract with premium $\bar{p}^i = p_i^{m_i} + B^i$ (surcharge $B^i \geq 0$) and add-on transfer $\bar{t}^i = \ell + A^i$ ($A^i \geq 0$) constructs these purely from observable parameters, implementing the price cap without requiring knowledge of the full type distribution.

Empirical specifications

This is a pure theory paper. There are no regression specifications or estimated equations. The illustration in Section I (p. 2552, Figure 1) uses calibrated parameters from Handel, Hendel, and Whinston (2015) to visualize the latent contract construction: CARA utility $u(z) = -e^{-\alpha z}/\alpha$ with risk-aversion coefficient $\alpha = 0.0004$, low risk $\underline{\theta} = 0.1$, high risk $\bar{\theta} = 0.2$. The figure shows indifference curves and the shaded region of contracts that attract the type $(H, \bar{\theta})$ but repel category $L$ consumers. These are calibration parameters used only for visual illustration; no data are fitted or tested.

Datasets used

Dataset	Role in paper	Wiki page
Handel, Hendel, and Whinston (2015) CARA calibration	Parameters for the illustrative Figure 1 only (not data in the usual sense: α, θ, θ̄ are fixed prior estimates)	No page yet

No empirical data are used in the main theoretical analysis or proofs.

When to read the full paper

Use the original if you are: designing or evaluating menu-based regulatory policies for insurance or other selection markets; studying delegation theory where an agent uses private data to screen principals; extending the framework to multiple firms, multidimensional types, or settings where regulators cannot impose purchase mandates; or working through the formal proofs and the online Appendix extensions (cream-skimming, constrained regulatory power, moral hazard). The six locators above point to the exact theorem and proposition pages.

Attribution and rights

Source: peer-reviewed, American Economic Review 113(10), 2023. No open-access licence was found in Crossref or OpenAlex metadata (access: closed). This page is a distillation extracted by an LLM on 2026-06-25 and is not human-verified or independently reproduced. Redistribution: extract-only.

Bhaskar, Dhruva, Andrew McClellan, and Evan Sadler. “Regulation Design in Insurance Markets.” American Economic Review 113, no. 10 (October 2023): 2546-2580. DOI: 10.1257/aer.20210710. © 2023 American Economic Association. Reproduced here in extract form only under fair-use scholarly commentary.