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.
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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,Footnote 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)Footnote 2; the non-Nash “anticipatory” equilibrium of Wilson (1977)Footnote 3; the “reactive” equilibrium of Riley (1979)Footnote 4; the equilibrium in a three-stage game examined by Hellwig (1987)Footnote 5; the two-stage game in which the informed agent moves first examined by Cho and Kreps (1987)Footnote 6; and the “weak” equilibrium with free entry characterized by Azevado and Gottlieb (2017).Footnote 7 There has been no clear consensus regarding either the appropriate definition of equilibrium or, consequently, of the allocations supported as equilibria.Footnote 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.Footnote 9
While equilibrium analysis in an insurance market with asymmetric information is plagued by multiplicity of equilibrium possibilities, Samuelson (1954, 1955)Footnote 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.”Footnote 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 dictatorFootnote 12 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.Footnote 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.Footnote 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.Footnote 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
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 costFootnote 16 of allocating the insurance contract C to an agent of type p is
which, given the definitions of \({W}_{L}\) and \({W}_{N}\), may be written as
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
which requires that the premiums collected cover the expected losses. The Paretian dictator’s problem may be written as
subject to the resource constraint (4) and the utility constraint on H-types
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.
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.Footnote 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).
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.
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.Footnote 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
which requires that each agent prefer the allocation that they are assigned over the alternative.Footnote 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.Footnote 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’.Footnote 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.
We may write the Paretian dictator’s efficiency problem as
subject to the resource constraint (8), the incentive constraints
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
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
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.Footnote 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.Footnote 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).Footnote 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.Footnote 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 modelFootnote 26 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 \({F}_{a}\le 0\),Footnote 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 dictatorFootnote 28 who can assign both s(x) and a may be written as one of maximizing the expected utility of the principal
subject a the constraint on the agents’ expected utility
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,Footnote 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.Footnote 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
which yields the first order condition
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,Footnote 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).Footnote 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.Footnote 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.Footnote 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.Footnote 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.Footnote 36
Data Availability
Not applicable.
Notes
Hoy restricts his analysis to the Wilson E2 equilibrium and the Spence-Miyazaki (which is referred to elsewhere in the literature as Miyazaki-Wilson-Spence, or MWS) equilibrium. There are, of course, many other possibilities, as touched upon below.
Rosenthal and Weiss characterize the equilibrium randomized strategies for n firms in the Rothschild and Stiglitz environment. This equilibrium is not immune to entry, however, as firm n + 1 could profitably enter using a different mixing strategy.
Wilson’s assumption is that firms considering a profitable defection anticipate that their offering would result in some existing contracts losing money and being withdrawn from the market, thereby making that defection unprofitable. This approach is addressed most recently in Mimra and Wambach (2019).
Riley’s assumption is that firms considering a profitable defection anticipate that their offering would spawn future offerings by other firms that would ultimately serve to make their initial defection unprofitable.
Hellwig characterized the sequential equilibrium in the game where at stage 1, the uninformed insurers offer contracts; at stage 2 the informed insureds choose from among the contract offers; and at stage 3 the uninformed insurers may reject the choices made in the second stage.
Cho and Kreps analyze a two-stage game in which the informed insureds move first by announcing a signal, then at stage 2, the uniformed insurers make contract offers contingent on those signals. Since this results in a continuum of sequential Nash equilibria, the “intuitive criterion” is applied as a refinement to obtain a unique outcome.
This is the approach followed in Levy and Veiga (2023).
One might argue, with some fairness, that there are also lots of possible equilibria in, say, the market for oranges. But observable structural differences in the market provide some guidance on the appropriate equilibrium. So for example if one observes large numbers of buyers and sellers, one might conclude that a price-taking competitive equilibrium would prevail, while if there were a single seller who set prices then a monopoly would be more appropriate. In contrast, with an insurance market, there are no structural markers that would lead one to conclude that the market participants use “Wilson foresight” or exhibit some other type of non-Nash behavior.
Because of this indeterminacy, Schmalensee (1984) noted that there is “no general theorem establishing that making information better [costlessly] always enhances efficiency in the [insurance market] context,” which is certainly the case when one applies an equilibrium analysis to the categorization problem. This statement is, however, incorrect in the context of an efficiency analysis, as I discuss below.
Musgrave (1985) describes these two short papers as “[c]arrying a benefit-page ratio without rival.”
Samuelson (1955), p. 350. Just as public goods entail externalities, the contracting problems posed by hidden knowledge generate externalites as well since better informed agents exploit their informational advantage.
The term Samuelson uses is “social planner,” but “Paretian dictator” is more usefully descriptive, and is the term that we shall apply going forward.
So, using the well-known Edgeworth Box analysis of an exchange economy, the Paretian dictator knows both parties’ indifference curves and has the ability to assign any allocation in the box. But the dictator cannot create resources and hence is constrained by the dimensions of the box.
The result in the Edgeworth Box setting is the “contract curve,” a concept well known to generations of undergraduate economics students.
One advantage of Samuelson’s approach is that it provides a clear separation between efficiency (maximizing the size of the pie) and distributional (how the pie is sliced) concerns. Once the efficient frontier is identified, a policy maker is free to choose the particular position on that frontier that accords with their distributional priorities. In contrast, the recent work by Einav and Finkelstein (2011) presents an alternative approach to defining efficiency in competitive insurance markets which, as noted by Rothschild (2024), is a concept in which an explicit indifference to distributional concerns is formally embedded.
The continuum assumption ensures, from the law of large numbers and the independence of the individual risks, that the expected resource cost is equal to the actual cost with probability one.
The utilities possibilities frontier depicted is for \({\overline{V} }^{H}\le V\left({p}^{L}, F\right)\). Higher values of \({\overline{V} }^{H}\) would result in over-insurance for H-types (Crocker and Snow 1985a, b), who would therefore have the incentive to induce the loss and profit from the resulting over-indemnification. Given this obvious moral hazard problem, we will focus on efficient contracts that result in under-insurance.
One of the most important tools in normative analysis is the use of hypothetical compensation tests to compare different regimes and to determine whether the gainers from the movement to a new regime can compensate the losers. Schmalensee’s (1984) observation that hidden information “rules out [such] compensation even in principle” is, however, not correct. The informational asymmetry simply places constraints on the allowable nature of the hypothetical compensation through the incentive constraint (7).
In practice, these incentive constraints are implemented in a couple of different ways. One approach (and that is implicit in our discussion here) is that the Paretian dictator offers to the agents contracts that satisfy the incentive constraints and the agents then reveal their type by the contract that they voluntarily select. In such a setting (7) is often referred to as a “self-selection” constraint. In other settings, particularly those with a continuum of types (see, for example, Crocker and Morgan (1998) where agents have private information about the magnitude their actual losses and can falsify claims), it is more convenient analytically to use a “direct revelation mechanism” of the type discussed by Myerson (1979) in which the agents send to the Paretian dictator a message about their type after which they are assigned contracts that satisfy (7) and therefore induce the truthful revelation of type. In such settings, (7) is often referred to as the “truth-telling” constraint (see footnote 32, below).
The dashed line depicts solutions in which the utility constraint (6) is forced to hold with equality. Since these solutions are Pareto dominated by point M, at a solution to the constrained optimization, constraint (6) would be slack for any \({\overline{V} }^{H}\) less than the H-type utility associated with M.
Note that any movement of the endowment along the locus EF is a state-contingent reallocation that satisfies the resource constraint (4). Put differently, the Paretian dictator imposes a zero-profit pooling insurance contract that moves the insureds from E to E’. Moreover, the isocost contours \({\overline{\pi }}^{H}\) and \({\overline{\pi }}^{L}\) depicted are the zero-profit loci (relative to the new endowment E’) for allocations assigned to the H- and L-types, respectively.
The intuition is that after being classified into group A, it may be possible to adjust the contracts assigned to them in a fashion that reduces their resource cost while making them no worse off. These resource savings can then be transferred to those in group B, making them strictly better off.
Put differently, if the utility constraint on H-types is forced to be satisfied with equality, a solution to the Paretian dictator’s efficiency problem for \({\overline{V} }^{H}\) set sufficiently low would result in a point on the hidden knowledge utility possibilities frontier located on the dashed segment below M in Fig. 2. Both types could then be made better off by moving to point M on the frontier, so at a solution the utility of the H-type would be greater than the reservation level \({\overline{V} }^{H}\) and therefore we would have \(\delta =0\).
Of course, since deductibles are used to screen agent types within each group after classification, the utilities possibilities frontier associated with classification continues must lie inside of that associated with symmetric information, as the latter entails full insurance for all agents.
More precisely, when the Rothschild and Stiglitz allocations {H*, A} are not supportable as a Nash equilibrium, the associated utilities are not on the utilities possibilities frontier FM depicted in Fig. 2. The Wilson E2 allocation, defined as the pooling allocation most preferred by the low-risk individual, is not on that frontier either, although the Spence-Miyazaki allocation is and results in the utilities position M. Hence, Hoy’s equilibrium analysis compares an allocation which is not on the frontier (the Rothschild and Stiglitz allocation) to either one that is not on the frontier either (Wilson E2) or one that is (Spence-Miyazaki).
Higher values of a result in a first order stochastically dominant shift of the probability distribution function to the right.
The point of reference in the typical principal-and-agent problem is that of the principal designing a contract to incentivize the agent to take a desired action. This is, however, exactly the problem faced by a Paretian dictator: To maximize the utility of the principal, subject to a utility constraint on the agent and the resource constraints of the economy (and, ultimately, any informational constraints that may arise).
An equilibrium approach in this environment would presumably involve a group of principals facing a group of agents, with the former trying to attract the latter by designing attractive contracts, which is precisely the situation in insurance markets. But that is not what is going on here, as the principal-and-agent literature is completely devoid of any appeal to equilibrium reasoning. Now, sometimes modelers may try to finesse this problem by assuming that the equilibrium involves the principal making a “take it or leave it” offer to the agent. But that is simply normative analysis without the label. To the extent that equilibrium analysis is designed to tell us something about what we observe in the real world (price markups by grocery stores are low because the market is competitive, markups by Microsoft in the late 90s were large because they had a 95% market share and were a monopoly), it is hard to figure out what real-world economy the “take it or leave it” equilibrium is supposed to represent. Perhaps some type of authoritarian regime.
Since the optimal sharing rule s(x) is a function, the solution approach is to apply either calculus of variations or control theory.
More precisely, the optimal action is a solution to \({\int }_{\underline{x}}^{\overline{x} }[G+U]{f}_{a}dx=h^{\prime}\).
An insurance analogue to this approach would be a setting in which the insured took a hidden action that affected the probability distribution of losses, which when realized were hidden knowledge known only by the insured, but who can invest in generating a (potentially falsified) claim that is observable.
Since we are dealing with a continuum of types, the implementation of the incentive constraint (7) differs from the (self-selection) approach in the two-types world. Begin by decomposing the function s(R) into a reporting function R(x) and a transfer function s(x), and let the preferences of an x-type agent be denoted as V(s,R|x). Following Myerson (1979) we consider as a solution technique the direct revelation mechanism in which the agent sends a message about her type, \(\widehat{x}\), and is then assigned the contractual allocation \(\{s\left(\widehat{x}\right), R\left(\widehat{x}\right)\}\). This allocation satisfies the incentive (now the “truth-telling”) constraint as long as \(V\left(s\left(x\right),R\left(x\right)|x\right)\ge V\left(s\left(\widehat{x}\right),R\left(\widehat{x}\right)|x\right)\) for every \(x,\widehat{x}\in [\underline{x},\overline{x }]\), which yields the first order condition for the revelation problem \(\frac{d}{d\widehat{x}}V\left(s\left(\widehat{x}\right),R\left(\widehat{x}\right)|x\right)=0\) when \(\widehat{x}=x.\) This is the constraint on the s and R functions that the Paretian dictator must respect in order for the incentive (truth-telling) constraint to be satisfied. Once the optimal reporting and transfer functions are identified, the bonus function can be recovered by observing that S(R) = S(x(R)) where x(R) is the inverse function. For specific details of the solution technique, see Crocker and Slemrod (2007).
A similar problem is examined by Crocker and Letizia (2014) in the design of an optimal payment for product returns between a manufacturer and a retailer. In this setting the manager takes an action that effects the probability distribution of product returns from consumers to retailers. The actual number of returns is, however, privately known by the retailer, who may engage in falsification to misrepresent the reported returns observed by the manufacturer. Here, the Paretian dictator’s optimization entails both delegation and incentive constraints, as in the earnings management setting.
Note that in this setting there is only one type of customer so there is no need for a utility constraint as in the case where there are two types.
And, as noted earlier, analyses in settings with hidden actions are devoid of market equilibrium discussions.
Using for the moment the Rothschild and Stiglitz competitive (Nash) equilibrium as a point of reference, the use of compulsory insurance as first suggested by Dahlby (1981) can support any informationally constrained Pareto optimal contract as an equilibrium. This can be easily demonstrated by reference to Fig. 1, where all agents are required to purchase the (zero profit) insurance policy that moves them from E to E’. Then the agents are permitted to purchase supplemental policies in the competitive market, which results in the equilibrium allocations (H, L’) that are incentive compatible and each of which earns zero profit off of those purchasing their policies. A similar result is obtained by Crocker and Snow (1985b) through the use of a balanced-budget tax and subsidy scheme applied to insurance contracts.
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Acknowledgements
I would like to thank Michal Hoy for publishing his article on risk classification in 1982 that planted the seed from which ultimately grew a productive research program. I would also like to thank Arthur Snow for being a critical part of that journey, and for teaching me welfare economics along the way.
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Crocker, K.J. The role of normative analysis in markets with hidden knowledge and hidden actions. Geneva Risk Insur Rev 49, 163–180 (2024). https://doi.org/10.1057/s10713-024-00101-z
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DOI: https://doi.org/10.1057/s10713-024-00101-z