Monopolistic Insurance and the Value of Information

The value of information regarding risk class for a monopoly insurer and its customers is examined in both symmetric and asymmetric information environments. A monopolist always prefers contracting with uninformed customers as this maximizes the rent extracted under symmetric information while also avoiding the cost of adverse selection when information is held asymmetrically. Although customers are indifferent to symmetric information when they are initially uninformed, they prefer contracting with hidden knowledge rather than symmetric information since the monopoly responds to adverse selection by sharing gains from trade with high-risk customers when low risks are predominant in the insurance pool. However, utilitarian social welfare is highest when customers are uninformed, and is higher when information is symmetric rather than asymmetric.


Introduction
Will a risk-neutral monopolistic insurer provide information to uninformed insurance applicants regarding their membership in either of two risk classes when that information is costless to obtain and transmit?The answer is no.The most rent the insurer can extract from a customer is the consumer surplus gained with full-and-fair insurance, which is measured by the Arrow-Pratt risk premium.Since the risk premium inherits concavity in the loss probability from concavity of utility in wealth, Jensen's inequality implies that profit is strictly greater at the average probability than at any mean-preserving combination of high and low probabilities.Indeed, the monopoly has an incentive to confuse informed

Insurance Contracting under Monopoly
The insurance market consists of a large number of insurance applicants who demand coverage for independent risks of losing an amount of wealth x.Each applicant is endowed with wealth w, the risk-averse utility function u(w), and a probability of incurring the loss x.The insurance pool consists of two classes of these individual risks, a proportion ) 1 , 0 (   of customers being endowed with probability H p of incurring the loss, the remaining proportion being endowed with probability . (Chade and Schlee [3] analyze adverse selection when a monopolistic insurer contracts with customers from many risk classes.Jeleva and Villeneuve [4] analyze monopolistic insurance in the two-type case when customers maximize rank-dependent utility.Asheim and Nilssen [5] examine the case when renegotiation is possible after customers initially choose contracts.)Uninformed customers believe they have the population's average risk, when evaluating a given contract.(That is, uninformed customers share an unbiased belief about risk class and, since expected utility is linear in probabilities, the value of a given contract is based on the average risk.By contrast, uninformed customers evaluating an informational regime in which they are perfectly categorized confront a classification risk when each customer's contract, as well as the probability of loss, depends on the customer's revealed risk class.)Informed customers know their risk classes, and this information constitutes their hidden knowledge when information is asymmetric.(With asymmetric information, customers are assumed to be better-informed than the insurer.Villeneuve [6] analyzes the opposite case, when the insurer is better-informed than its customers.Customers are also assumed to be identical except for the risk of loss.Landsberger and Meilijson [7] analyze monopolistic insurance when customers possess hidden knowledge of both risk and risk aversion.)Applicants treat all these parameters as exogenous, along with the insurance contracts on offer.
The insurer is monopolistic and can make all-or-nothing contractual offers consisting of a premium m and an amount of coverage c, and can make more than one such offer.The expected utility of a t-type applicant opting for a contract with premium m and coverage c is denoted by A contract ] , [ c m must satisfy the voluntary participation constraint if it is to attract a type t customer.When information is asymmetric, contracts must be incentivecompatible, with the members of each risk class preferring their intended contract.When customers are uninformed, the value of the contract ] , [ c m is given by the expected value of Equation (2), Following Malinvaud's [8] lead, the law of large numbers is assumed to apply to each risk class individually, so that a contract [m t , c t ] taken exclusively by those in risk class t earns the insurer a per-customer profit equal to the contract's expected value, with probability one.As the aggregate risk presented by each risk class is negligible, the insurer behaves as if neutral to risk, that is, as if there were no aggregate risk.(As shown by Malinvaud [8], and discussed in more general terms by Rothschild and Stiglitz [1], individual risks are "socially removed by the operation of the law of large numbers.")Hence, the insurer maximizes expected profit, treating the parameters describing the applicant pool as exogenous.
One way to account for the insurer's profit is to assume that the monopolist is a Pareto-relevant agent whose welfare is measured by monopoly profit, and each applicant's wealth is then where o w is exogenous.An alternative approach assumes that ownership rights in the monopoly are uniformly distributed among the insurance applicants, so that the wealth w taken as exogenous by applicants and by the monopolist is actually endogenous, being given by where is monopoly profit per applicant.Since the risk to this profit is negligible, applicants behave as if wealth w is riskless.(In this environment, the monopolist is a mutual or participating insurer.As in Picard's [9] analysis of competitive participating insurers facing adverse selection, in equilibrium customers receive neither dividends nor calls for additional premiums, there being no aggregate risk.)In either case, when evaluating alternative information regimes, the monopolist and applicants treat w as if it were exogenous.

Symmetric Information
Stiglitz [2] shows that when information is symmetric, the monopolistic insurer can attain perfect rent extraction, providing full coverage while charging a premium that extracts all consumer surplus, the latter being measured by the Arrow-Pratt risk premium, ) , ( w p t Proof Definition (6) for the risk premium implies , which is negative if and only if u′′(w) < 0. ■ Intuitively, with a given state-contingent wealth (w, w − x), as p increases from zero to one, the risk premium at first increases and then decreases due to the concavity of u, and thus inherits concavity in p.It follows that the monopolist is averse to risk about probability, and Jensen's inequality implies establishing the monopolist's strict preference for uninformed customers.Proposition 1: Informing applicants of their risk classes is strictly unprofitable for a monopoly insurer taking the wealth w of applicants as given.
After being informed of risk class, H-types are worse off and L-types are better off, but from an ex ante perspective, they are indifferent between informed and uninformed contracting since they obtain no surplus in either regime and are neutral to risk about probability.(Schlesinger and Venezian [11] analyze a risk-neutral, monopolistic insurer who can manipulate the probability of loss to maximize expected profit.Customers benefit from the monopolist's choice of probability if their risks are initially greater.By contrast, in the present context, H-types benefit and L-types lose when their risks are changed to p , but applicants evaluate these changes with ex ante expected utility.)Proposition 2: Applicants not knowing their risk classes and taking wealth w as given are indifferent between symmetrically informed and symmetrically uninformed contracting.
Proof The expected utility of an uninformed customer is An informed customer's utility is so that each applicant's ex ante evaluation of symmetrically informed contracting is the expected utility It follows that applicants are indifferent between symmetric information and uninformed contracting.■ Proposition 1 implies that a monopolistic insurer has an incentive to confuse informed applicants and convince them of being average.Since Proposition 2 indicates that applicants are indifferent to such manipulation, the result could be an increase in social welfare.This conjecture is confirmed for utilitarian social welfare in Section 5.
In sum, a monopolistic insurer prefers symmetrically uninformed over symmetrically informed contracting in order to extract the greatest rent from the applicant pool, while applicants are indifferent between these two contracting regimes.The situation is rather different when insurance contracting is wholly competitive.Insurers are then indifferent to their customers' information status as expected profit is always zero, while customers prefer uninformed contracting, inasmuch as symmetric information exposes them to classification risk.Specifically, although neutral to risk about probability, applicants are averse to bearing the risk about state-contingent wealth that accompanies risk classification under competitive contracting.(In the symmetrically uninformed regime, each applicant hastheaveragerisk and competitive equilibrium provides full-and-fair insurance for this risk of loss.In the symmetrically informed regime, class-specific full-and-fair insurance is provided in equilibrium, and L-types enjoy an increase in welfare while H-types suffer a decline.Since marginal utility is diminishing, this classification risk reduces an applicant's ex ante evaluation of symmetrically informed contracting below that of the symmetrically uninformed regime.See Crocker and Snow [12] for further analysis of this case.)By contrast, with monopolistic insurance, each applicant's contract provides the same expected utility as the null contract, and classification introduces only risk about probability, towards which applicants are risk neutral.

Asymmetric Information
When customers possess hidden knowledge of risk class, a monopolist's viable contractual offers must meet not only the voluntary participation constraint in Equation (3) for both types, but must also satisfy incentive compatibility constraints, ensuring that each type t prefers its intended contract.An optimal contractual offering satisfies these constraints while maximizing expected profit per customer, The incentive constraints in Equation ( 8) impose a cost on the monopoly insurer, unless customers are uninformed.(Landsberger and Meilijson [13] show that when many loss amounts are possible, the incentive constraints in Equation ( 8) do not impose a cost and the monopoly can attain first-best rent extraction if there are some loss amounts that can be incurred only by H-types.)Hence, a monopoly prefers uninformed customers both to avoid the cost of adverse selection, and to then maximize the rent extracted from them.
Stiglitz [2] characterizes the monopolistic equilibrium for this environment, showing that (i) pooling contracts are not profit-maximizing; (ii) only the H-type incentive constraint is binding; (iii) the H-type contract provides full coverage; and (iv) the voluntary participation constraint for L-types is binding.
consisting of the null contract, chosen by L-types, and the full-coverage contract that extracts maximum consumer surplus from H-types.Observe that the pair Equation ( 9) satisfies the voluntary participation and incentive constraints in Equations ( 3) and ( 8), and earns the insurer a positive expected profit.In contrast, with *    , the profit-maximizing pair has the form where the premium ) ( L L c m extracts maximum rent from L-types given coverage L c .The L-type contract in the pair at Equation (10) provides some coverage for L-types while maintaining strict equality in their participation constraint and sacrificing profit 0 on the H-type contract.
Proposition 3: When *    , high-risk applicants, taking wealth w as given, prefer asymmetric over symmetric information.
Proof Given wealth w, when information is symmetric, the expected utility of H-types is . When information is asymmetric and *    , their expected utility is A more general assessment of the interests of insurance applicants is possible by adopting the veil-of-ignorance approach developed by Harsanyi [14,15] in which applicants evaluate alternative information regimes without hidden knowledge of risk class.
Proposition 4: When *    , applicants choosing from behind a veil of ignorance regarding risk class and taking wealth w as given prefer contracting with asymmetric rather than symmetric information.
Proof With asymmetric information, expected utility is for L-types, and for the asymmetric information regime, where the inequality follows from the assumption * both equal zero for all values of  , and each applicant's expected utility is Hence, applicants' expected utility is higher when contracting with asymmetric rather than symmetric information.■ The following is an immediate consequence of Proposition 4. Corollary 1: Applicants choosing from behind a veil of ignorance regarding both risk class and the proportion of high risks prefer contracting with asymmetric rather than symmetric information.
When information is asymmetric, the monopolist avoids the cost of adverse selection if applicants are uninformed, and its interests are thus in conflict with those of its customers indicated in Propositions 3 and 4. Matters are again different under competitive insurance, where insurers are indifferent between informed and uninformed customers.(Dahlby [16] explores conditions under which L-types obtain greater coverage under monopolistic contracting than under competitive contracting.)Under competition, customers prefer to remain uninformed, even if classification risk is insurable, since adverse selection imposes an additional cost.

The Social Value of Information
In evaluating informed vs. uniformed contracting under symmetric information, the monopolist prefers uninformed contracting while its customers are indifferent.When evaluating contracting with symmetric rather than asymmetric information, the monopolist and its customers are in conflict.In both cases, one can ask "Which option better serves social welfare?" since the evaluations of the applicants and the monopolist are based on selfish interests and competitive behavior that presumes each applicant'swealthis wholly exogenous.
Since there is no aggregate risk, there are only two risk premiums relevant to the social welfare comparisons examined in the following sections under either approach to accounting for monopoly profit.One is an applicant's premium for classification risk relevant in the ex ante assessment of symmetrically informed contracting; the other is the L-type premium for bearing residual, uninsured risk under asymmetrically informed contracting.

The Social Value of Symmetric Information
First, consider the social value of symmetric information, which generates a per-customer monopoly profit of or o w .From the ex ante perspective, there is a representative applicant whose expected utility constitutes the measure of social welfare when applicants share the monopoly profit.When the monopolist is a Pareto-relevant agent, utilitarian social welfare is measured by the representative applicant's certainty-equivalent wealth plus the monopolist's profit.
Proposition 5: The social value of symmetric information is negative.Proof When applicants share monopoly profit, the representative applicant's expected utility with uninformed contracting is given by Under symmetric information, certainty-equivalent wealth for an H-type applicant is and for an L-type applicant is ) , ( Hence, the representative applicant's evaluation of symmetric information is where s  is the premium for classification risk, which is positive given risk aversion.Since Equation ( 12) exceeds Equation ( 15), the social value of symmetric information is negative.
When the monopolist is Pareto-relevant, the representative applicant's certainty-equivalent wealth , and social welfare is Under symmetric information, s w is equal to o w in Equation ( 11) for monopoly profit, as well as in Equations ( 13) and ( 14) for certainty-equivalent wealth, and the representative applicant's evaluation of symmetric information is , where s ˆ denotes the premium for classification risk when profit is not distributed to applicants.It follows from Equation ( 17) that social welfare is , which is less than Equation ( 16), implying that symmetric information has negative social value.■ Since the monopolist achieves full rent extraction under either symmetrically informed or symmetrically uninformed regimes, monopolistic contracting results in an interim efficient sharing of risk as riskaverse applicants receive full coverage.However, by the accounting of Proposition 5, social welfare is higher with symmetrically uninformed contracting, and thus social welfare increases when a monopolist can, at no cost in resources, successfully convince informed applicants that they are average.

The Social Value of Symmetric vs. Asymmetric Information
Under asymmetric information, profit is where a w equals either


, which is attained when  is greater than or equal to *  .
The following result shows that profit is higher when information is symmetric rather than asymmetric regardless of applicant preferences and the allocation of monopoly profit.
Lemma 2: Proof The difference in profit between symmetric information given in Equation ( 11) and asymmetric information given in Equation ( 18) is Since the left-hand side of Equation ( 21) is positive, if and only if the term within braces is negative, which requires that one or both of the derivatives within braces must be greater than one.However, differentiating Equation ( 6) for the risk premium with respect to wealth yields . Hence, regardless of wealth effects on the applicants' degrees of risk aversion, profit is greater under symmetric rather than asymmetric information when applicants share the monopoly profit.■ The difference in monopoly profit between the symmetric and asymmetric information regimes is mirrored in the social ranking of the two regimes.
Proposition 6: Social welfare is greater with symmetric rather than asymmetric information.
Proof For H-types, certainty-equivalent wealth under asymmetric information is and for L-types is ) , ( When applicants share monopoly profit, the difference in social welfare between symmetric and asymmetric information, obtained from Equations ( 13), ( 14), (22), and (23), is where the second equality uses Equations ( 11) and ( 18) for profit.Since Equation ( 24) is positive for all ) 1 , 0 (   , social welfare is higher with symmetric rather than asymmetric information when monopoly profit is shared by applicants. When applicants do not share the profit, s w is equal to o w in Equations ( 11), (13), and ( 14), and replaces w a in Equations ( 18), (22), and (23), so the difference in social welfare between symmetric and asymmetric information is since the monopolist is Pareto relevant.Again, social welfare is higher with symmetric information.■ The intuition for Proposition 6 is that monopoly profit either represents a transfer of wealth to a Pareto-relevant agent (the monopolist), which has no social value, either positive or negative, under utilitarian social welfare, or is distributed to applicants and, on average, just offsets the rent from them.The problem with asymmetric information is that profit is too low.With symmetric information, the monopolist offers interim efficient contracts, but with asymmetric information, adverse selection prevents the monopolist from achieving full rent extraction from L-type customers, and the contracts offered are, therefore, not interim efficient.
Finally, Koch's [17] analysis suggests an important caveat relevant to Proposition 6. Recognizing that, in a dynamic insurance context, both adverse selection and (re)classification risk can impose costs on the contractual relationship, Koch explores the interaction between these two market failures.Reporting simulation results for a model calibrated to US data, Koch demonstrates a Theory of the Second Best result by showing that eliminating one but not both market failures need not be Pareto improving.Although developed in the context of a (regulated) competitive environment, the same conclusion surely holds when insurance contracting is monopolistic.(In a related vein, Handel et al. [18] report simulation results suggesting that eliminating reclassification risk by banning its use, thereby artificially creating hidden knowledge, could nonetheless be welfare enhancing.Mahoney and Weyl [19] present simulation results investigating the tradeoff between market power and adverse selection, suggesting that greater market power can be socially valuable as an instrument for mitigating an adverse selection externality.)

Conclusions
Whether the informational environment is symmetric or asymmetric, a monopolistic insurer always prefers contracting with uninformed applicants, and would profit from sowing confusion among them regarding their risk classes when applicants are informed.Indeed, sowing confusion increases utilitarian social welfare when information is symmetric since applicants are indifferent between uninformed contracting and contracting with symmetric information.However, choosing from behind a veil of ignorance, applicants prefer asymmetric over symmetric information as H-types gain consumer surplus when L-types dominate the insurance pool.Nonetheless, social welfare is higher with symmetric information due to the cost of adverse selection reflected in the rent the monopolist fails to extract from L-types, while the social value of symmetric information is negative, since it exposes applicants to a classification risk that reduces their ex ante expected utility.
rent the monopolist fails to extract from each L-type applicant due to adverse selection.Note that ) Additionally, the L-type contract provides partial coverage with a positive premium if *