The role of normative analysis in markets with hidden knowledge and hidden actions

Welfare economics has largely disappeared from sight, a disappearance that is strange in the sense that economists have not ceased to make welfare statements. (Atkinson 2009)

Abstract

When examining the economic impact of a change in policy regime, one may elect to conduct a positive analysis that considers the effect of such a policy change on the allocations supported as a market equilibrium. An alternative approach would be to perform a normative analysis that examines the effect of the regime change on the set of Pareto optimal allocations in the economy. In settings where the Fundamental Welfare Theorems hold, the isomorphism between equilibrium and Pareto efficient allocations implies that the choice of approach is largely a matter of convenience. When this isomorphism fails to hold, as in settings with hidden information or hidden actions, the equilibrium allocations may not be efficient so that a normative analysis of the alternative efficiency frontiers is required to determine whether the policy change is desirable in the sense of permitting potential Pareto improvements. If so, then an equilibrium analysis may be used to determine the extent to which such improvements may be realized by market participants.

1 Introduction

In a thought-provoking article, Michael Hoy (1982) examined the desirability of permitting the use of categorical discrimination by insurers, that is, the practice of offering insurance applicants contracts tailored to their observable characteristics when these traits are known to insurers and are themselves imperfectly correlated with the underlying propensities for suffering losses, which is hidden knowledge that is known only to the insureds. Given that categorization is ubiquitous in insurance markets, and that those that are favorably classified into a lower risk group benefit from this practice while those who are less fortunate tend to be disadvantaged, this is an important and practical policy matter. To analyze the economic implications of categorization, Hoy considered the effect that the introduction of categorical discrimination had on insurance market equilibria,¹ concluding that the efficiency consequences of such classification are ambiguous and depend critically on the particular competitive behavior assumed to sustain an equilibrium in the market with hidden knowledge.

This non-result is a consequence of a more fundamental problem with the use of equilibrium analysis to understand the economic implications of alternative regulations on contracting when hidden knowledge is an important element. Since the influential work on informationally constrained insurance markets by Rothschild and Stiglitz (1976) indicating the potential non-existence of a pure strategy Nash equilibrium, there has arisen something of a cottage industry developing alternative equilibrium concepts attempting to solve the non-existence-of-equilibrium problem. A representative but non-exhaustive list would include the mixed strategy Nash equilibrium characterized by Rosenthal and Weiss (1984)²; the non-Nash “anticipatory” equilibrium of Wilson (1977)³; the “reactive” equilibrium of Riley (1979)⁴; the equilibrium in a three-stage game examined by Hellwig (1987)⁵; the two-stage game in which the informed agent moves first examined by Cho and Kreps (1987)⁶; and the “weak” equilibrium with free entry characterized by Azevado and Gottlieb (2017).⁷ There has been no clear consensus regarding either the appropriate definition of equilibrium or, consequently, of the allocations supported as equilibria.⁸ Lacking such a consensus, attempts to predict the economic effects of policies such as categorical discrimination by comparing equilibria with and without the practice are destined for failure.⁹

While equilibrium analysis in an insurance market with asymmetric information is plagued by multiplicity of equilibrium possibilities, Samuelson (1954, 1955)¹⁰ in his pathbreaking examination of the optimal provision of a public good faced the opposite problem: “[T]he fatal inability of any decentralized market or voting mechanism to attain or compute this optimum.”¹¹ Rather than relying on a market mechanism to determine the allocation of the public good, Samuelson introduced a fictitious entity that we shall refer to as the Paretian dictator¹² who is all powerful (so can assign by fiat consumption allocations), all knowing (therefore has complete knowledge of all preferences), is benign (in that the assignments are guided by the Pareto criterion) but is ultimately constrained by the resources of the economy.¹³ Moreover, Samuelson provides a convenient mathematical shortcut to characterize the solution to the Paretian dictator’s problem. In a two-agent economy, this boils down to the problem of maximizing the utility of one agent subject to a utility constraint of the other, and the resource constraints of the economy. By varying the level of the utility constraint, one can characterize all possible solutions to the dictator’s problem.¹⁴ This approach has provided the underlying foundation for normative analysis in economics, and we shall follow Samuelson’s lead in the discussion that follows.¹⁵

2 An insurance environment with two types of agents and symmetric information

As a point of departure, consider an economy with a continuum of risk-averse agents each of which has the von Neumann-Morgenstern utility function $U({W}_{i})$, with ${W}_{i}$ denoting an agent’s wealth in the loss (i = L) and the no-loss (i = N) states. Given the state-contingent wealth $W=({W}_{L}, {W}_{N})$, and the probability of loss p, an agent’s expected utility may be written as

$$V\left( {p, W} \right) = \left( {1 - p} \right)U\left( {W_{N} } \right) + pU\left( {W_{L} } \right).$$

(1)

The agents differ only in terms of their probability of suffering the loss, L, with a proportion λ suffering the loss with probability ${p}^{H}$ and the proportion (1 − λ) suffering the loss with probability ${p}^{L}$, where ${p}^{L}< {p}^{H}.$ For the purposes of this section, agent type (H or L) is observable. An insurance contract $C\equiv (m, I)$ is composed of a premium m and indemnity, I, paid in the event of the loss L. Assuming that each agent possesses the initial wealth $\overline{W}$, we have ${W}_{N}= \overline{W}-m$ and ${W}_{L}= \overline{W}-m-L+I$. The resource cost¹⁶ of allocating the insurance contract C to an agent of type p is

$$\pi \left( {p, C} \right) = m - pI$$

(2)

which, given the definitions of ${W}_{L}$ and ${W}_{N}$, may be written as

$$\pi \left( {p, W} \right) = \overline{W} - \left( {1 - p} \right)W_{N} - pW_{L} - pL$$

(3)

where $W\equiv \left({W}_{N}, {W}_{L}\right).$

Because type is observable, the Paretian dictator can treat the agents differently and assign the wealth allocation ${W}^{i}\equiv ({W}_{N}^{i}, {W}_{L}^{i})$ to a type-i agent as long as the allocations satisfy the resource constraint of the economy, which is written as

$$\lambda \pi \left({p}^{H}, {W}^{H}\right)+\left(1-\lambda \right)\pi \left({p}^{L}, {W}^{L}\right)\ge 0$$

(4)

which requires that the premiums collected cover the expected losses. The Paretian dictator’s problem may be written as

$$\begin{array}{c}Max\\ \{{W}^{H}, {W}^{L}\}\end{array} V({p}^{L}, {W}^{L})$$

(5)

subject to the resource constraint (4) and the utility constraint on H-types

$$V\left({p}^{H}, {W}^{H}\right) \ge {\overline{V} }^{H}.$$

(6)

The solution to this constrained optimization problem is illustrated in Fig. 1 as the wealth allocations depicted as H and L. The initial endowment of the agents is depicted as E, and the locus labeled as $\overline{\pi }$ which has the slope $-\frac{(1-\overline{p })}{\overline{p} }$ where $\overline{p }= \lambda {p}^{H}+(1-\lambda ){p}^{L}$ depicts the wealth allocations (including F and E’) that satisfy the resource constraint with equality when both types are assigned the same contract. The loci labeled ${\overline{\pi }}^{i}$ with slopes $-\frac{(1-{p}^{i})}{{p}^{i}}$ are the isocost curves depicting the allocations where $\overline{\pi} \left({p}^{i}, {W}^{i}\right)= {\overline{\pi }}^{i}$. The point denoted H is the allocation that minimizes the resource cost ${\overline{\pi }}^{H}$ of maintaining the utility level ${\overline{V} }^{H}$ for the type-H agents. Then any wealth allocation for L-types along the isocost locus ${\overline{\pi }}^{L}$, when paired with H for H-types, satisfies the resource constraint (4) with equality, and so the wealth allocation depicted as L maximizes the expected utility of the L-types.

Fig. 1

Solution to the Paretian dictator’s problem

By varying the level of reservation utility ${\overline{V} }^{H}$ in (6), all of the possible solutions to the Paretian dictator’s problem may be characterized. This results in the symmetric information utilities possibilities frontier depicted in Fig. 2, the concave shape of which reflects the agents’ risk aversion.¹⁷ To see this, let ${W}^{H}$ and ${W}^{L}$ denote full insurance allocations for H- and L-types, respectively. Then the resource constraint (4) may be written as $\overline{W }- \lambda {W}^{H}-\left(1-\lambda \right){W}^{L}- \overline{p }L=0$, and the utility possibilities frontier may be represented as the parametric function $\left\{{V}^{H}\left({W}^{H}\right), {V}^{L}\left({W}^{H}\right)\right\}$ where ${V}^{H} \equiv U({W}^{H})$ and ${V}^{L} \equiv U(\frac{\overline{W }- \lambda {W}^{H}- \overline{p}L }{1-\lambda })$. The derivative of the parametric function is given by $d{V}^{H}/d{V}^{L}=\frac{d{V}^{H}/d{W}^{H}}{d{V}^{L}/d{W}^{H}}= -\frac{U{\prime}({W}^{H})(1-\lambda )}{U{\prime}({W}^{L})\lambda }$. At the F point, ${W}^{H}= {W}^{L}$ and the slope of the utilities possibilities frontier reduces to $-\frac{1-\lambda }{\lambda }$. For ${V}^{H}< {V}^{L}$ (which implies ${W}^{H}< {W}^{L}$), the ratio $\frac{U{\prime}({W}^{H})}{U{\prime}({W}^{L})}$ > 1 and is increasing as ${W}^{H}$ (${W}^{L})$ decreases (increases).

Fig. 2

Utilities possibilities frontiers

Finally, note that the Fundamental Theorems of Welfare Economics hold in this setting, as indeed they must. When firms compete in the offering of contracts to attract customers, the market segments into contracts conditional on the observable types. Following Rothschild and Stiglitz (1976), the competitive (Nash) equilibria are the zero-profit and full insurance contracts {H*, L*} depicted in Fig. 3, as there are no profitable defections that can attract H- (L-) types from H* (L*), and the allocations supported in equilibrium are also Pareto optimal. Moreover, any Pareto optimal contract can be supported as an equilibrium allocation after a suitable redistribution of the initial endowments. To see this, note in Fig. 1 that if the initial endowment is changed from E to E’, which satisfies the resource constraint (4), then the zero-profit market equilibria relative to this new endowment are the Pareto optimal allocations {H, L}. Thus, the isomorphism between Pareto optimal allocations and market equilibria is preserved, although the two classes of contracts are the result of distinctly different processes: the Paretian dictator assigning allocations by fiat in the former, while firms are competing in their contractual offerings to attract customers in the latter.

Fig. 3

Competitive Nash equilibrium

3 Normative analysis with hidden knowledge

Now consider the environment of hidden knowledge examined by Rothschild and Stiglitz (1976) in which only the agents know their probabilities of suffering the insurable loss L. We will assume that the Paretian dictator is constrained by the same informational asymmetry that constrains the market participants, so that its ability to treat the agents differently in the assignment of allocations based upon a priori knowledge of their risk types has been eliminated. But this does not imply that the Paretian dictator cannot treat the agents differently based upon their underlying types, only that it faces constraints in its ability to do so.¹⁸ More precisely, in order to assign the allocations ${W}^{H}$ and ${W}^{L}$ to H- and L-types, respectively, the allocations must satisfy the incentive constraint

$$V\left( {p^{i} , W^{i} } \right) \ge V\left( {p^{i} , W^{j} } \right)\,{\text{for}}\,i, j \in \left\{ {H, L} \right\}$$

(7)

which requires that each agent prefer the allocation that they are assigned over the alternative.¹⁹ The solutions to the problem of maximizing (5) subject to (4), (6) and (7) identify the class of informationally constrained Pareto optimal allocations.

The implications of the incentive constraint can be appreciated with reference to Fig. 1. The symmetric information optimal allocations {H, L} cannot be assigned to the agents by the Paretian dictator since both types of agents prefer L to H. Holding the H-types allocation at H, the best that the Paretian dictator can do for the L-types is to assign them the allocation L’, which results in a lower level of expected utility for that agent than would the allocation L.

Figure 2 depicts the utilities possibilities frontier with hidden knowledge as lying everywhere inside of that associated with symmetric information except at the allocation F where both types are assigned the same allocation so that the incentive constraint (7) is satisfied trivially.²⁰ Accordingly, if one were able to wave a magic wand to eliminate the informational asymmetry, then movement from the regime of hidden knowledge to the symmetric information regime would constitute a potential Pareto improvement in the sense of the Samuelson (1950) criterion: The utility frontier with symmetric information lies everywhere on or outside the frontier with hidden knowledge. Hence, any benefits realized from the change in informational regime can, in principle, be redistributed via lump-sum transfers along the symmetric information frontier to a utility distribution that Pareto dominates any initial distribution in the hidden knowledge frontier.

The candidate for competitive (Nash) equilibrium identified by Rothschild and Stiglitz (1976) is the Pareto dominant member of the set of informationally consistent allocations, defined as those allocations satisfying both the incentive constraint (7) and the constraint that each contract earns zero profit, which is depicted as {H*, A} in Fig. 3. The equilibrium non-existence problem arises because, for λ below a critical value that we will denote as $\widehat{\lambda }$, the utilities associated with these allocations are located on the dashed line below M in Fig. 2 so that there exist allocations that satisfy both the resource constraint (4) and the incentive constraints (7), and which are Pareto superior to {H*, A}. But these allocations also entail a cross-subsidy in which profits made off the L-types are used to cover losses incurred on the H-types and so cannot be sustainable as a Nash equilibrium as an insurer could profitably defect by simply dropping the money-losing contract.

Even so, a version of the Fundamental Welfare Theorems holds in this environment. As shown by Crocker and Snow (1985a, b), the competitive Nash equilibrium, if it exists, is also an informationally constrained Pareto optimal allocation. Further, with reference to Fig. 1, any informationally constrained Pareto optimal allocation such as {H, L’} can be supported as a Nash equilibrium after a redistribution of the initial endowment from E to E’.²¹

4 Risk classification

Now consider the question addressed by Hoy (1982) concerning the desirability of risk classification, only viewed from a normative perspective as in Crocker and Snow (1986). The informational environment continues to be one where agents possess hidden knowledge of risk class, but they can now be divided into two groups, denoted A and B. Group membership is observable to the market participants and, therefore, to the Paretian dictator. There are still two types of agents (H and L) but the groups are assumed to have different proportions of the two types. Letting ${\lambda }_{i}$ denote the proportion of H-types in group i, we assume that $0 <{\lambda }_{A}< {\lambda }_{B}<1$ so that categorization based upon group membership is imperfectly informative.

Since group membership (A and B) is observable, the Paretian dictator can assign different wealth allocations to each group but, to treat the members within each group differently based on their unobservable underlying type (H or L), the allocations would have to satisfy an incentive constraint. Specifically, denoting as $\left({A}^{H}, {A}^{L}\right)$ and $\left({B}^{H}, {B}^{L}\right)$ the allocations assigned to the A and B groups, respectively, and assuming that the proportion $\theta$ of the population belongs to group A, the resource constraint is written as.

$$\theta \left[ {\lambda_{A} \pi \left( {p^{H} ,{ }A^{H} } \right) + \left( {1 - \lambda_{A} } \right)\pi \left( {p^{L} ,{ }A^{L} } \right)} \right] + \left( {1 - \theta } \right)\left[ {\lambda_{B} \pi \left( {p^{H} ,{ }B^{H} } \right) + \left( {1 - \lambda_{B} } \right)\pi \left( {p^{L} ,{ }B^{L} } \right)} \right]{ } \ge 0$$

(8)

We may write the Paretian dictator’s efficiency problem as

$$\begin{array}{c}Max\\ \{{A}^{H}, {A}^{L}, {B}^{H},{B}^{L}\}\end{array} V({p}^{L}, {B}^{L})$$

(9)

subject to the resource constraint (8), the incentive constraints

$$V\left( {p^{i} , A^{i} } \right) \ge V\left( {p^{i} , A^{j} } \right)\,{\text{for}}\,i, j \in \left\{ {H, L} \right\}\,\,{\text{and}}\,V\left( {p^{H} , B^{H} } \right) \ge V\left( {p^{H} , B^{L} } \right)$$

(10)

and a set of utility constraints on both types in group A and the H-types in group B. Letting $\left({W}^{H}, {W}^{L}\right)$ denote a solution to the Paretian dictator’s problem in the absence of risk classification, we will write these utility constraints as

$$V\left( {p^{i} , A^{i} } \right) \ge V\left( {p^{i} , W^{i} } \right)\,for\,i \in \left\{ {H,L} \right\}\,and\,V\left( {p^{H} ,B^{H} } \right) \ge V\left( {p^{H} ,W^{H} } \right).$$

(11)

Given that $\left({W}^{H}, {W}^{L}\right)$ is a feasible solution to the classification problem, these utility constraints guarantee that any solution to the classification problem that differs from that of the no-classification problem is necessarily Pareto superior. Crocker and Snow (1986) demonstrate that these solutions differ when

$$\frac{\delta }{{\mu }^{H}}<\frac{\lambda -{\lambda }_{A}}{{\lambda }_{A}(1-\lambda )}$$

(12)

where $\delta$ and ${\mu }^{H}$ are the Lagrange multipliers associated with the H-type utility and incentive constraints, respectively, in the no-classification constrained optimization problem.²² If the utility level ${\overline{V} }^{H}$ for H-types in that problem is set equal to that of the allocation F, then both types of agents are assigned the same contract and the incentive constraint does not bind, so ${\mu }^{H}=0$ and (12) can never hold. If ${\overline{V} }^{H}$ is set sufficiently low, then one can show that $\delta =0$ and then (12) would always hold.²³ Moreover, as ${\lambda }_{A}$ declines, categorization becomes more informative and, when ${\lambda }_{A}$ is small enough, inequality (12) holds indicating that categorization is potentially welfare enhancing. In the limit, ${\lambda }_{A}=0$ and classification is perfectly informative and so (12) always holds. Finally, if the classification is completely uninformative, so ${\lambda }_{A}=\lambda$, then (12) could never hold. We may therefore conclude that the utilities possibilities frontier associated with classification must coincide with the no-classification frontier at the point F, and either coincides with or lies outside of the no-classification frontier elsewhere. Accordingly, moving from the no-classification regime to the classification regime would constitute a potential Pareto improvement, in the sense of Samuelson (1950).²⁴

In contrast to the ambiguous results obtained from the equilibrium approach, a proper normative analysis definitively concludes that, in this hidden information environment, it is welfare-improving to permit insurers to classify their customers based upon observable characteristics that are imperfectly correlated with their underlying risks.²⁵

These normative results are of more than merely theoretical interest, as Finkelstein et al. (2009) use this framework to examine the efficiency and distributional effects of restricting gender-based annuity pricing in the United Kingdom. From the perspective of an insurer selling annuities, a high-risk customer is one who lives a long time and therefore collects a lot of annuity checks, and a low-risk customer is one who dies earlier. Assuming that longevity is hidden information known only to the customer, an insurer can treat the customer types differently by offering two types of annuity payout, one that is front-loaded with higher payments earlier that decline over time (which is preferred by low-risks who don’t expect to be around later) and another with a level payment (which is fine with high-risks since they expect to survive to collect the larger checks in future). Moreover, the observable genders of the annuity customers may serve as a useful classification tool since the male (female) category contains a lower (higher) proportion of the high-risk types.

Using mortality data, Finkelstein et al. use maximum likelihood estimation to calibrate the model which is then applied to calculate the welfare effects of gender-based annuity pricing. With reference to the no-classification (hidden information) utilities possibilities frontier depicted in Fig. 2, they calculate that the welfare gain of permitting gender-based classification at F would be zero since at that point the utilities possibilities frontiers of both the classification and the no-classifications coincide. But the welfare gains from permitting gender classification increase as one moves southeast along the utilities possibilities frontier and is maximized at point M where the welfare benefit of permitting gender-based classification is calculated to range from 0.018 to 0.025% of retirement wealth, depending on the customers’ degree of risk aversion.

5 Normative analysis with hidden actions

The canonical hidden action model²⁶ is that examined by Holmstrom (1979) which involves a risk-averse principal and a risk-averse agent who share an observable monetary payoff, x, which is a continuous variable on the interval $[\underline{x},\overline{x }]$ with the probability distribution function F and density f. Denoting the share of the payoff accruing to the agent as s(x), we may write the utility of the principal as G(x-s(x)) and that of the agent as U(s(x)), where G and U are strictly concave utility functions. The goal is to design a sharing rule, s(x), that optimally shares the risk of the random monetary payoff, x, between the principal and the agent.

The agent may take an action, a, that makes larger payoffs more likely, so ${F}_{a}\le 0$,²⁷ but the agent receives disutility, h(a), from higher levels of that action. If the action, a, were observable to all, then the symmetric information problem facing a Paretian dictator²⁸ who can assign both s(x) and a may be written as one of maximizing the expected utility of the principal

$$\begin{array}{c}Max\\ \{s(x), a\}\end{array} \underset{\underline{x}}{\overset{\overline{x}}{\int }}G\left(x-s\left(x\right)\right)f\left(x|a\right)dx$$

(13)

subject a the constraint on the agents’ expected utility

$${\int }_{\underline{x}}^{\overline{x}}\left[U\left(s\left(x\right)\right)-h\left(a\right)\right]f\left(x|a\right)dx\ge \overline{H }$$

(14)

where $\overline{H }$ denotes the agent’s reservation level of utility. The resource constraint requires that the agent’s share, s(x), and the principal’s share, x − s(x), add up to the total payoff, x. A solution to this problem is a sharing rule s(x) that results in the efficient risk sharing of the risk associated with the random payoff between the risk-averse principal and the risk-averse agent,²⁹ and the assignment of an action that balances the utility benefits of a larger payoff and the agent’s disutility from choosing higher actions.³⁰

Now, a hidden action is introduced by assuming that the action, a, chosen by the agent is not observable to anyone else, including the Paretian dictator, although the dictator can still observe the actual payoff, x, and assign the sharing rule s(x). Given this assignment, the agent selects the action that solves the maximization problem

$$\begin{array}{c}Max\\ a\end{array} {\int }_{\underline{x}}^{\overline{x}}\left[U\left(s\left(x\right)\right)-h\left(a\right)\right]f\left(x|a\right)dx$$

(15)

which yields the first order condition

$${\int }_{\underline{x}}^{\overline{x} }[U(s(x)){f}_{a}-h^{\prime}f]dx=0.$$

(16)

With the hidden action, the informationally constrained Paretian dictator’s constrained optimization problem is (13) subject to (14) and the delegation constraint (16) that assigns to the agent the selection of the action a. Even though the Paretian dictator cannot assign the hidden action, it can influence the agent’s choice of that action through the sharing rule. Accordingly, the optimal sharing rule in the presence of a hidden action reflects a tradeoff between optimal risk sharing, on the one hand, and incentivizing the agent to supply a higher level of the privately costly action, on the other.

An instructive example of the interplay between a hidden action and hidden knowledge, albeit in a non-insurance setting,³¹ is provided by the model of earnings management examined by Crocker and Slemrod (2007). The environment consists of a risk neutral principal (firm) who contracts with a risk neutral agent (manager). The agent takes a privately costly action, a, that effects the distribution of the payoff (earnings), x, as in Holmstrom, but in this setting the payoff when realized is hidden knowledge known only to the agent. As a result, this is a setting where there is both hidden knowledge (regarding x) as well as a hidden action (regarding a).

While the Paretian dictator cannot observe x, it can observe a report generated by the agent, denoted as R, which may differ from the actual payoff if the agent engages in falsification. The cost to the agent of reporting a payoff of R when the actual payoff is x can be denoted as g(R-x) where g(0) = 0 and g’, g” > 0. The task of the Paretian dictator is to design the optimal bonus schedule s(R) that stipulates the payment to the agent, s, when the observed earnings report is R. In this setting, the utility of the principal is written as x-s(R) while that of the agent is s(R)-g(R-x)-h(a).

The Paretian dictator’s optimization problem maximizes the expected utility of the principal subject to an expected utility constraint of the agent, as well as delegation and incentive constraints analogous to (16) and (7).³² The optimal bonus schedule reflects a tradeoff between the incentives generated by the hidden knowledge and those associated with the hidden action. A flat bonus schedule (s’(R) = 0) has the advantage of providing the agent with no incentive to generate a higher report through falsification since bearing the cost of falsifying would not increase the payment to the agent. But the disadvantage of the flat bonus schedule is that it provides the agent with no incentive to take the privately costly action. Alternatively, a steeper bonus schedule (s’(R) > 0) provides the agent the incentive to take the costly action, but also generates a positive return to falsification. The optimal bonus schedule, therefore, reflects a balancing of these two effects.³³

A final example of the Paretian dictator at work is provided by Bourgeon and Picard (2014) who examine the role of auditing to deter fraudulent claiming by insurance purchasers. An insurance contract is a specification of a premium, P, and an indemnity, I, to insure the potential loss L. Whether the loss occurs or not is hidden knowledge known only by the claimant who may choose with probability $\alpha$ to file a claim even if a loss has not occurred. Claims may be audited by incurring a cost, c, that reveals to the insurer whether the claim is legitimate or not. Letting $\beta$ denote the probability of an audit, the assumption is that non-audited claims are paid in full, audited claims that are found to be fraudulent are not reimbursed at all, and audited claims found to be legitimate are underpaid. In particular, when a claim is filed the insurer observes the random variable $\widetilde{x}$ which represents the upper “legal” limit on the proportion of the claim that can be underpaid, and the insurer decides on the amount it will underpay $z\left(\widetilde{x}\right) \left(\le \widetilde{x} \right).$ As a result, legitimate claims receive the indemnity $\left(1-z(x)\right)I$ where $z(x)$ is the amount of insurer “nitpicking.”

There are three informational scenarios examined that result in different constraints on the Paretian dictator. The first-best (full information) is when the Paretian dictator selects the insurance contract terms $\{P, I, z\}$, as well as $\alpha$ and $\beta$, and is a solution to the problem of maximizing the expected utility of the insured subject to the constraint that premiums equal the expected indemnity payments.³⁴ The first-best solution entails $\alpha =\beta =0$ so there is no fraud or monitoring, I = L so there is full insurance, and z = 0, and there is not nitpicking. The second-best (informationally constrained) is when the Paretian dictator assigns the insurance contract terms and $\beta$, but delegates to the insured the selection of $\alpha$ so that there is now a delegation constraint in the dictator’s problem. The solution to the second-best problem entails the assignment of a $\beta$ that completely deters fraud, so $\alpha$ = 0, and there is no nitpicking so z = 0 as well. The third-best (even more informationally constrained) occurs when the Paretian dictator can only assign the contract terms $\{P, I, z\}$ and delegates to the insurer and insured the selection of $\alpha$ and $\beta$ who make their choices in a simultaneous move game. The solution to this optimization problem involves some nitpicking, so $z(x)>0.$

6 Discussion

While normative analysis is an important and powerful tool for policy analysis, it unfortunately provides no predictions regarding market equilibrium outcomes, although one might expect that the market participants would have a strong incentive to find a way to achieve a result that is located on the efficiency frontier. This proclivity is illustrated by the wide range of tools that have been adopted by insurers to deal with the informational asymmetries that they face.

When insurance purchasers possess hidden knowledge about their loss probabilities, were insurers to charge a single premium for all their customers then the Akerlof (1970) “lemons” problem could manifest itself through an adverse selection “death spiral” in which the market collapses and only the highest risks are served. The use of perfectly informative risk classification, if it were available, would permit the tailoring of premiums to individual risks and therefore would mitigate this market failure. If the risk classification were imperfectly informative, there may be subgroups where the customers have hidden knowledge regarding their types which would make them susceptible to market failure, but then insurers could screen their customers with deductibles in the usual fashion. And, as was demonstrated earlier, further risk classification of the type first examined by Hoy (1982) permits Pareto improvements.

There are other classification tools that are commonly implemented by insurers and that have been shown to permit Pareto improvements. Bond and Crocker (1991) demonstrate that permitting insurers to classify their customer based upon their observable consumption of goods that are either correlated with underlying risks (such as giving premium discounts to students with good grades or customers with exemplary credit scores) or affect those risks directly (such as charging higher premiums to customers who smoke) is potentially Pareto-improving. In a similar vein, Crocker and Snow (2011) demonstrate that permitting insurers to classify their customers based upon the nature of the peril causing the loss, as when insureds are charged different deductibles depending on whether their house was destroyed by a fire, a flood, or a hurricane, is desirable because such an approach reduces the resource cost of screening by the use of deductibles. Finally, Crocker and Zhu (2021) examine the classification of insureds based upon their willingness to submit to a test that is imperfectly informative of their risk types, such as having drivers agree to install a telematic device that records their driving behaviors or when an applicant for life insurance agrees to take a physical examination, and conclude that to permit this type of voluntary classification is efficient.

7 Conclusions

When the Fundamental Theorems of Welfare Economics hold, policy analysis can be performed in an equilibrium setting because of the isomorphism between Pareto optimal allocations and those supported as equilibria. Unfortunately, in insurance settings with hidden knowledge, there is substantial debate regarding the appropriate equilibrium concept.³⁵ Moreover, some of these candidates support allocations as equilibria that are Pareto dominated even under the informational constraints that characterize the market, a vaguely troubling outcome in competitive markets with free entry. Accordingly, policy analysis in an informationally constrained setting requires the formal application of normative analysis to characterize and to compare the utilities possibilities fronters associated with alternative policy regimes. In this fashion, one may ascertain whether changes in a policy regime permit potential Pareto improvements in the sense defined by Samuelson (1950). Whether such improvements may be attained in an equilibrium setting remains an open question, although there are tools that policymakers may use to enable the realization of such gains.³⁶