Primary care delivery, risk pooling and economic efficiency

Abstract

The consequences of consumer-driven health care under different health insurance plans are studied by means of a game theoretic approach. Suitable demand-side cost-sharing can induce consumer behavior that avoids over-treatment when there are information asymmetries between providers and consumers, leading to the efficient recommendations and provision of treatment by providers. If under-treatment can be penalized, then a full insurance model that pays providers a fixed salary and fee-for-service or one that requires patients to present a referral letter before specialist care is delivered also achieves provision efficiency. The two models, however, yield higher welfare for consumers. Hence, the findings in this paper favor some amount of regulation in health-care markets.

Introduction

Health systems are composed of delivery and financing systems. The primary objective of a health-care financing system is to protect a population from health-related financial risk. In most developed countries, a third party pays for health care and may or may not observe the deliverables of the services rendered. In many cases, financial contributions of individuals to a health-care system are raised by a third party in the form of taxes or a health insurance premium, which may or may not be proportional to the income earned. The financing arrangement often creates incentives that affect both the quantity and quality of the delivery of health services.¹

Less obvious is the fact that the manner by which health care is financed may also influence the way a medical market is organized.² Specifically, this paper studies the effects of different health insurance plans on the industrial organization of a consumer-driven health-care service market and their subsequent impacts on health-care provision. In particular, the importance of information asymmetries in linking market organization and health-care provision is emphasized throughout this paper.

To demonstrate these points, this paper provides a stylized model of prepaid health care consisting of a third party payer, a health-care provision sector and a consumer sector. The third party, which may be a government, a sickness fund or a collection of sickness funds, a social security agency or regulated competitive private insurers, provides health-care insurance to the enrolled consumers. The independent service provision market consists of a fixed number of service providers, who will choose their professions as either general practitioners or specialists differentiated by the scope of services they provide. The consumer sector consists of consumers who seek consultation and treatment when they fall ill. A variety of supply-side payments and demand-side cost-sharing arrangements, as well as their corresponding consequences of delivery and purchase, are discussed and analyzed. The setting described above corresponds to a medical market that consists of general practitioners and specialists where patients have their first point of contact with health care, and hence this is defined as a primary care market.³

Uncertainty and information in the health-care market are emphasized throughout the paper. As Arrow [1] stresses, medical insurance improves consumer’s well-being under uncertainty. Furthermore, under information asymmetry, providers tend to observe the health condition non-observable to consumers, thus creating the chances of provision inefficiency either in the form of over- or under-treatment, as investigated in the literature of credence goods surveyed by Dullock and Kerschbamer [9]. The problem of asymmetric information also exists between the third party payer and the service providers. Although a third party may observe the exact tangible treatment, there may be non-observable treatment costs, especially related to the effort and time spent on the case by providers in the absence of patients. And yet, these non-observable costs, and thus non-contractible effort, must be compensated to induce appropriate services. This information asymmetry thus provides a chance for service providers to overcharge or demand a higher payment from the third party payer. Chalkley and Malcomson [5] have a detailed account of the implementation of the first-best treatment qualities and non-observable effort when health care is purchased by a welfare-maximizing government. The contractual arrangement relies on salary payment, capitation and supply-side cost-sharing. Chalkley and Khalil [4] compare outcome-based and traditional treatment-based payment for alleviation of the payer-supplier agency problems, where the former payment arrangement is more applicable for less severe illnesses. Siciliani [24] considers payment contracts based on activity level and waiting time in publicly funded health-care systems where providers have superior information and may dump patients. When there is asymmetric information, payment of information rent may become necessary to induce efficient provision.⁴

Besides supply-side demand inducement, the problem of demand-side moral hazard is another important issue in the presence of insurance. Pauly [18] and Pauly and Ramsey [20] describe the problem of demand-side moral hazard as consumers demanding a higher quantity of treatment because of insurance coverage than otherwise if prices are set at marginal costs.⁵ In the analysis by Pauly and Ramsey [20], consumers are assumed to know about their conditions, and they simply tell providers what to provide. Similarly, in the analysis of Blomqvist and Leger [3], patients actively decide the quantity of treatment. Although these modeling strategies reflect realism in some circumstances, many consumers exercise their decision power only to a very limited extent. In particular, the participation of consumers in treatment decisions largely relies on the information about the illness conditions provided by the providers based on their professional judgments.⁶ In many cases, the decisions of consumers may only be left with a passive binary choice of either acceptance or rejection of treatment recommended.

In the presence of these information problems, the organization of the medical market affects the provision of health-care services. As described above, supply-side demand inducement is modeled as a combination of provision of excessive (i.e., unnecessary) treatment and stinting in a provider’s efforts. Meanwhile, the problem of demand-side moral hazard is modeled as acceptance of potentially unnecessary treatments. Hence, supply-side inducement is modeled as an active decision of providers, while demand-side moral hazard is modeled as a passive choice of consumers.⁷ Economic efficiency in the provision of health services is, correspondingly, defined as the provision of appropriate treatment in the absence of stinting of provider’s efforts. Using these modeling strategies, this paper addresses both the problems of supply-side demand inducement and the demand-side moral hazard simultaneously.⁸ These features thus capture the relationship among health insurance, purchasing, and provision decisions in the medical sector.

To accomplish the objectives mentioned above, the analysis below adopts a game theoretic approach by incorporating health insurance in a search model developed by Wolinsky [26]. Correspondingly, consumer’s choice to either accept or reject treatment has potential to become a means to monitor the performance of providers. By refusing to receive treatment and by seeking for an alternative provider, a consumer may induce an appropriate recommendation from a provider. Hence, such consumer behavior may become a disciplinary device for deterring a provider’s misbehavior. However, the consumer’s incentives for engaging in such behavior may directly depend on how the underlying health insurance contract held by the patient is structured and on what alternative providers are available in the market. This paper shows that appropriate treatment without stinting can be induced under the appropriate choice of insurance parameters. By increasing the out-of-pocket cost of receiving care from a specialist, who mainly specializes in provision of sophisticated treatments, the third party payer can induce a market organization in which consumers consult general practitioners first as gatekeepers before visiting specialists, avoiding provision of unnecessary treatments, and thus achieving efficient coordination of care. Such gatekeeper systems can be formed by voluntary consumer behavior driven by a suitable insurance design. In contrast, when full insurance is offered, demand inducement and stinting of providers’ effort are unlikely to be avoided since the incentives for doctor searching are halted, leading to specialist-dominated markets and inefficient provision. To provide full insurance to consumers but avoid over-treatment and stinting, a salary payment model that pays information rent to induce economically efficient provision is considered. To further avoid the payment of information rent, an arrangement requiring providers to make a referral for specialized treatment is analyzed. Finally, consumer well-being under the different arrangements is compared. It is shown that both contracting and transfer of purchasing power to a third party from consumer welfare dominate a non-sophisticated insurance arrangement for achieving efficient rationing of health services, hence, arguing for a sophisticated and regulated health-care system.

The rest of the paper is organized as follows. Second section describes “The model”. Third section studies the relationship between cost sharing and the health-care market from an industrial organizational perspective. The section also investigates alternative measures under which full insurance is offered and economically efficient treatments are provided. Fourth section reports the conditional ordering of the different financing arrangements discussed in the previous section. Finally, the article presents with the “Conclusion” in the fifth section.

The model

Overview

Consider consumers who may be healthy or ill. All consumers are enrolled in an insurance policy offered by a third party, which is described by a premium and a set of co-insurance rates as well as a set of rules to be followed by the enrollees. If consumers fall ill, they are unable to self-diagnose and self-treat. Consequently, they visit health service providers. The number of service providers is exogenously fixed, and their job is to diagnose and prescribe treatments. Prices for treatment are posted by services providers and are observable. Meanwhile, consumers decide whether to receive a treatment or not. If a consumer does not agree to receive a treatment, the consumer searches for another service provider for a treatment. If the consumer agrees, then she pays out-of-pocket the co-payment for the treatment calculated and based on the observable prices. The doctor searching process ends when a consumer receives treatment. Once treatment is received, a patient is cured if providers do not under-provide. The following subsections describe in detail the characteristics of the consumers and providers, illness, medical technology, information structure, the third party as well as the strategies of the third party, consumers and providers.

Consumers and service providers

Consumers are ex ante identical and risk-averse; they may be healthy or ill. There is only one illness, which occurs with probability, ρ, with two conditions: severe and mild, as in Pauly and Ramsey [20] and Chalkley and Khalil [4]. The illnesses considered are the less visible and acute ones, such as an infection, that allow consumers to refuse immediate treatments. The probability of contracting a mild illness is given by ω ∈ [0,1], which is common knowledge to the third party, the providers and the consumers. Consumers earn income, w; consumption, c, is the income after payment of health-care expenditures, such as payment of the health insurance premium regardless of health conditions and the out-of-pocket spending for health care if ill. Consumer’s preference over consumption is given by a concave utility function, U(•), satisfying U′(c) > 0 and U″(c) > 0.

Only providers who recommend and render treatments can discern the condition of an illness. The responsibility of a provider is to diagnose the health conditions of a consumer and to decide which treatment to recommend and render, either light or heavy treatment, corresponding to the two types of illness, which will be further elaborated below.⁹ Providers aim to maximize economic profit.¹⁰ Doctor searching behavior and treatment-purchasing strategies of consumers, as well as pricing and servicing strategies of providers, are described in “Strategies of the providers and consumers.” Before that, the following section describes the technology of treatment and the structure of information that gives rise to asymmetry between providers and consumers or the third party alike.

Treatment technology and information structure

Treatments of diseases are composed of a bundle of tangible health services and treatment, as well as non-tangible treatment efforts spent by the service provider. Corresponding to the two illness conditions, there are two appropriate levels of treatment: heavy treatment for severe conditions and light treatment for mild conditions. The heavy treatments are made up of observable specialized treatment and imperfectly observed service provider’s efforts, e _H (i.e., even though patients may observe the efforts exerted by providers, it is hard to judge the quality of those efforts by patients and the third party). Heavy treatment may or may not involve hospital-based care. Similarly, the light treatments are made up of observable general treatment and imperfectly observed efforts of the provider, e _L . The monetary costs of light and heavy treatment are L and H. It is assumed that H > L and e _H  > e _L . The total economic cost of the heavy treatment becomes H + e _H , while the total economic cost of the light treatment is L + e _L , assuming the monetary cost of effort is unity. Without loss of generality, let e _L  = 0 and e _H  = e. Furthermore, the values of e, L and H are common knowledge among the providers, consumers and the third party. However, only the costs of specialized and general treatment are verifiable, but the effort cost, e, which is imperfectly observable, is assumed to be non-verifiable to both the consumers and the third party. Both specialized and general treatment can treat an illness of mild condition, but only heavy treatment (i.e., specialized treatment plus adequate effort) can treat an illness of severe condition. Such medical technology and information structure open the possibility of provision of unnecessary treatment and stinting of effort.

Strategies of the providers and consumers

On the supply side, service providers decide which treatment to render. The classification of a provider is determined by his capacity and willingness to supply treatments. A provider who provides only light treatment is called a general practitioner. A provider who has the capacity to provide heavy treatment is called a specialist. Specialists are assumed to have the capacity to provide light treatment, but general practitioners cannot provide heavy treatment. Specialization is assumed to be costless. This assumption is indeed unrealistic since specialization involves non-market idiosyncratic and market training costs; for example, general practitioners may receive training in family medicine, while specialists receive further training in different specialties. However, the assumption enables us to develop a theory apart from the segregation of medical professions that rely on costly training.

Providers charge prices, P _L and P _H, satisfying P _L  ≥ 0 and P _H  ≥ 0, respectively, for the light and heavy treatments, which are observable to the third party and to consumers.¹¹ Since the supply prices are the minimum prices needed to induce services, the pair of prices (P _L , P _H ) also indicates the specialization of a service provider. For example, a pair of prices (P _L , P _H  = ∞) indicates that the service provider is a generalist who refuses provision of heavy treatment, and a pair of prices (P _L  = ∞, P _H ) indicates that the service provider is a specialist who refuses provision of light treatments. Meanwhile, (P _L , P _H ), satisfying ∞ > P _L  ≥ 0 and ∞ > P _H  ≥ 0, indicates a specialist who provides both treatments. Certainly, since the cost of heavy treatment exceeds the cost of light treatments, it implies that the supply prices must satisfy P _L  < P _H . It is assumed that the providers will commit to the posted prices charged for the treatments.

The service provider must also decide what treatment to recommend. To treat a mild illness condition, a provider can choose to deliver the observable heavy tangible treatment but exert low effort. Since consumers cannot tell whether they need heavy or light tangible treatments, they need to judge based on health outcomes after treatments are received. Observing health outcomes, however, does not help because both types of treatment cure an illness of mild condition. Therefore, the provider can induce demand for more expensive heavy tangible treatment and shirk in his effort. To treat a severe condition, however, a provider cannot under-treat a consumer without being noted because health outcomes are ultimately observable. Under-treatment is assumed to be subject to market discipline or disciplinary actions of a regulatory authority. Such action may be a temporary suspension of a license if the consumer files a complaint against the provider and the provider is proven to have practiced under-treatment, causing a continuation of an illness condition.¹² Alternatively, the provider may be controlled by the Hippocratic oath of a physician. A study of under-treatment in the presence of insurance and bankruptcy constraints such that effective penalties are infeasible shall be left for future research.

Stinting may occur because of diagnostic mistakes made by providers or may be simply due to ambiguity in diagnostic results. In the environment described above, if there is no segregation of professions in the medical market, then providers will tend to prescribe heavy treatment if they are not sure of the illness conditions, because heavy treatment cures illnesses of both conditions. If the profession of service providers is segregated, then there will be more referrals of false-positive severe illness conditions from general practitioners to specialists to avoid market punishment because of under-treatment by mistake. Both scenarios create more opportunities for demand inducement, even though some of this inducement may be unintentional. To simplify the analysis without loss of substance, it is assumed that providers can correctly diagnose the appropriate level of treatments needed.

Formally, a provider makes a pricing decision for the two types of treatments, P ∈ R ²₊ , that is also his/her specialization decision, S, as described above, and a decision on the treatment recommended (i.e., either light or heavy treatments), R. The specialization decision is made based on the consumer’s strategy and the insurance policy provided by the third party. The recommendation decision is made based on a set of diagnostic results, D. The provider, however, may randomize by choosing a probability of recommending a heavy treatment, y, such that y: D → [0,1]. If y = 0, then a light treatment is recommended. Given the information structure and penalty system, y = 1 when a severe condition illness is found; otherwise, practice of under-treatment (i.e., y = 0) is penalized. However, when an illness of mild condition is found, a service provider may choose y ∈ [0,1]. Hence, the provider may claim the need for heavy treatment when, in fact, only light treatment is needed. In such cases, specialized treatments are rendered without any treatment effort.

The payoff of a provider is now described. The economic profit made by a provider is P _J –J–e _J , J = H, L. There are two cases. If a provider honestly provides heavy treatment when they are needed, y = 1, then his payoff is:

$$ P_{H} -H-e. $$

(1)

However, if the provider induces demand when a light treatment is needed, then his payoff becomes

$$ P_{H} -H. $$

(2)

Hence, he essentially pockets e.

In this model, only heavy treatment providers can induce demand, while under-treatment by light treatment providers is subject to market discipline via penalties imposed by a professional regulatory body. In reality, general practitioners also induce demand. To incorporate this opportunity, false subjective illness signals must be added to the model. Hence, a consumer may perceive himself to be sick, but in fact he is healthy. Then, a healthy consumer who thinks he is sick may receive unnecessary treatment from general practitioners. The extent of demand inducement by general practitioners will then depend on the proportion of consumers who belong in such a category. This feature can be embedded in the model by expending the definition of illness with mild conditions to include the false-positive cases. To simplify our analysis, the prevalence of false self-perceived sick cases is ignored. It follows that if the providers were to provide light treatments, then they would provide them honestly, yielding a payoff of P _L −L.

On the demand side, when consumers fall ill, they visit providers for diagnosis and treatments. Consumers choose either a general practitioner or a specialist to visit. They then decide whether to accept treatment based on the treatment recommended by the provider at the price charged and on the number of previous doctor visits made. Hence, the decision of a consumer after their choice of a provider is a couple of probabilities of accepting treatments, x: S × B×O × R×T → [0,1]², where S is the specialization of a provider, B is the subjective beliefs of contracting a severe illness that maps ω and y into [0,1], O is the out-of-pocket expenditures on the treatment recommended (i.e., the product of the coinsurance rate and price charged by a provider for the recommended treatments, ϕ _J P _J , J = H, L), R is a set of treatment recommendations, and T is the set of positive integers (i.e., the number of previous doctor visits made). Correspondingly, the probability of continuation of searching by the consumer is given by 1 − x.

Furthermore, before the consumer visits a provider, he already suffers from the illness, incurring a disutility k on each visit. The magnitude of disutility, k, is identical between severe and mild illness conditions so that a consumer cannot distinguish illness conditions based on subjective judgment.¹³ If he accepts the treatment offered on his first visit, then his payoff becomes

$$ U\left( {w-\alpha -\phi_{J} P_{J} } \right)-k $$

(3)

where α is the premium paid for insurance coverage. If he rejects treatment, then he can search for another provider for recommendation. If he decides to accept the recommendation on his second visit, then his payoff becomes

$$ U\left( {w-\alpha -\phi_{J} P_{J} } \right)-2k. $$

(4)

If he accepts treatment on his nth visit, then his payoff is U(w−α−ϕ _J P _J )−nk. Hence, the consumer suffers less if he receives treatment earlier. If he is healthy, then his payoff equals U(w−α).

Third party payer

The third party payer is modeled as an insurance provider. An insurance policy is defined as a contractual triple, I = {α, ϕ _L , ϕ _H }, which consists of a premium and a pair of coinsurance rates. The insurance provider offers an actuarially fair insurance in the absence of loading cost, whose objective is assumed to design an insurance policy that maximizes the consumer welfare, anticipating the reactions of consumers and providers.¹⁴ This assumption is plausible under a benevolent social health insurance model. However, in the private health insurance model, it is necessary to assume insurance providers maximize profit in a perfectly competitive market such that equilibrium economic profit is zero and consumer’s welfare is maximized via the process of competition among health insurance providers. In a tax-based model, it is comprehended as the government allocating merely sufficient money to cover the expected health-care cost such that there is no surplus left and charging a set of user fees. However, the set of measures that influences both the behavior of consumers and providers available to a third party, M, can be much richer than simply a pair of coinsurance rates.¹⁵ The strategy-dependent actually fair insurance premium under a breakeven condition is given by

$$ \alpha = \rho \left[ {\omega \left( {\left( {1 - y} \right)\left( {1 - \phi_{L} } \right)L + yx\left( {1 - \phi_{H} } \right)\left( {H + e} \right)} \right) + \left( {1 - \omega } \right)\left( {1 - \phi_{H} } \right)\left( {H + e} \right)} \right] $$

(5)

where α is the premium paid to the third party ϕ _L , ϕ _H and is the coinsurance rate for the light and heavy treatments, respectively. Hence, (5) suggests that the premium raised must be equal to the expected payout for the costs of treatment rendered. It is assumed that the third party does not explicitly negotiate prices with the providers in the competitive provision market but may influence the pricing decision using some administrative measures.¹⁶ Also, since effort is non-contractible, reimbursement is executed according to the observable treatment provided.

Equilibrium

The equilibrium concept used in this paper is a Bayes–Nash equilibrium in which the service providers select their recommendation strategies by observing the true type of diseases; the consumers update beliefs on the likelihood of an illness condition, based on the recommendation strategies of the service providers and the probabilities of the two illness conditions. And then, they decide whether to reject or accept a treatment recommendation, and thus continue searching for another provider or do not. However, the strategies of the service providers and consumers must be made based on the terms of an insurance policy offered by the third party. The generic configuration of the equilibrium of the model is a set of strategies, P, S, x* and y*, which satisfy the following conditions:

For any level of ω ∈ [0,1], k > 0, and a set of measures, τ, such that τ ∈ M,

• (E1) The service providers choose y* ∈ [0,1] subject to x* ∈ [0,1] chosen by consumers to maximize economic profit.

• (E2) The consumers choose x* ∈ [0,1] subject to y* ∈ [0,1] chosen by service providers to maximize the expected utility.

• (E3) P ∈ [P _L ,P _H ] such that P _L is either L or ∞ and P _H is either H + e or ∞, assuming either price competition or the third party only reimburses at marginal cost if insurance coverage is offered (i.e., ϕ _J  ≠ 1, J = H and L).

• (E4) An equilibrium specialization strategy, S: 1 − x* → P

• (E5) $\alpha * = \rho \left( {\left( {\omega \left( {1 - y * } \right)\left( {1 - \phi_{L} } \right)L + y * x * \left( {1 - \phi_{H} } \right)\left( {H + e} \right)} \right) + \left( {1 - \omega } \right)\left( {1 - \phi_{H} } \right)\left( {H + e} \right)} \right)$

(E1) and (E2) follow the standard notion of a mixed strategy Nash equilibrium. (E3) depicts the pricing decision of service providers under price competition among providers for consumers or the reimbursement rule of the third party. (E4) is the equilibrium strategy of specialization that depends on the searching strategy of the consumer. (E5) is the payout for expected medical costs given the strategies of the service providers and consumers (i.e., the equilibrium premium).

Different equilibria, in terms of specialization and provision recommendations by service providers and acceptance strategies of consumers, may emerge, all of which are related to the terms of the insurance policy offered and thus the policy measures used by the third party. Consequently, any behavioral outcomes will be formed relative to the design of an insurance policy.¹⁷ Due to the fact that the set of policy measures available to the third party, M, can be very rich and is not restricted only to the set of coinsurance rates, the number of possible insurance-dependent equilibria is likely to be large. A full analysis of all the mixed equilibria is certainly beyond the scope of this paper. Among all possible equilibria, there are only two generic types of pure strategy equilibrium associated with the simple form of insurance policy considered in this paper: a pooling and separating equilibrium. The pooling (i.e., specialist) equilibrium is the one in which all the providers become specialists, providing either only heavy treatment or both treatments. In the separating equilibrium, also known as the mixed equilibrium below, service providers separate themselves into general practitioners who provide only light treatments and specialists who provide only heavy treatments. A set of equilibria among all the possible ones that feature the characteristics of the mechanisms used by a third party in the presence of information asymmetry are analyzed in “Equilibrium”.

The analyses below adopt the following steps: first, specify a configuration; second, prove that the configuration is a pure strategy Nash equilibrium; third, identify an insurance policy that supports the configuration. The intermediate steps as to how the configuration is arrived at, however, are not explored. It is convenient to consider ω that lies between 0 and 1 throughout the rest of “Equilibrium.” The limiting cases when ω is equal to either zero or one are then explored in Proposition 5 in “Ranking.”

Mixed-provider model

A mixed-provider model has the following configurations:

• (S1) Providers segregate into general practitioners and specialists. General practitioners produce light treatment exclusively and price at P _L  = L and P _H  = ∞. Specialists produce only heavy treatment exclusively and price at P _L  = ∞ and P _H  = H + e. They honestly recommend treatment needed by the consumer (i.e., y = 0 if light treatments are needed and y = 1 if heavy treatments are needed).

• (S2) Consumers visit general practitioners first. If light treatment is recommended, then consumers accept treatment. If heavy treatments are recommended by a general practitioner, then consumers would reject treatment and visit a specialist (i.e., x = 1 if light treatments are recommended on the first visit to a generalist; otherwise, x = 0). If heavy treatments are recommended by a specialist, then consumers accept (i.e., x = 1 on the second visit to a specialist who recommends heavy treatments) the recommendation.

• (S3) The third party offers a consumer welfare maximizing actuarially fair insurance policy, I ^s = {α ^s, ϕ _L  = 0, ϕ ^s _H  ∈ (0,1]}}, where the subscript s stands for segregation, supporting (S1) and (S2).

It is necessary to show that (S1) and (S2) are pure Nash equilibrium strategies; thus, given searching strategy (S2) of the consumers, the recommendation strategy of the providers (S1) holds, and vice versa. Hence, it is necessary to show that providers have no incentives to provide both types of treatment by deviating from (S1), given (S2). Suppose there is a deviant provider who was a general practitioner or a specialist and now supplies both treatments and posts a price schedule (P _L , P _H ). Thus, the deviant provider plans to threaten other providers by undercutting price and stealing consumers. The deviant provider, however, cannot make any positive economic profit by charging P _L  > L, because he cannot attract consumers. If he charges P _L lower than L, then he suffers an economic loss. Similarly, the logic holds for selecting P _H , resulting in P _H  = H + e. Therefore, by providing both treatments, he must charge at (P _L  = L, P _H  = H + e). If the consumers follow (S2), the provider will never provide heavy treatment, even though they are recommended, if only light treatment is needed, because according to (S2), consumers will search for other providers. Hence, if the heavy treatment recommendation is accepted by consumers, then treatments are needed. Thus, for the deviant provider, providing both treatments is equivalent to providing only the light treatment or heavy treatment since she makes zero economic profit anyway. Consequently, providers have no incentives to provide both light and heavy treatments. Given (S1), it is necessary to check under what features of the insurance policy would support (S2).

To support strategy (S1), it is necessary to find an insurance contract that supports the searching strategy adopted by the consumers, (S2). Now consider the expected payoff of a consumer if (S2) is followed,

$$ \begin{aligned} V^{s} = \left( {1 - \rho } \right)u\left( {w - \alpha^{s} } \right) + \rho \left[ {\omega u\left( {w - \alpha^{s} - \phi_{L} P_{L} } \right) + \left( {1 - \omega } \right)u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right)} \right] \\ - \rho \left[ {\omega k + \left( { 1- \omega } \right) 2k} \right] \\ \end{aligned} $$

(6)

Given the strategy of the providers, (S1), the consumers evaluate the cost and benefit of visiting a general practitioner first. If (S2) is followed, then the expected payoff must yield higher benefit than visiting a specialist first. Hence,

$$ \omega u\left( {w - \alpha^{s} - \phi_{L} P_{L} } \right) + \left( {1 - \omega } \right)u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right) - \omega k - \left( {1 - \omega } \right)2k \ge u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right) - k. $$

(7)

Conditions (6) and (7) imply

$$ u\left( {w - \alpha^{s} - \phi_{L} P_{L} } \right) - u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right) \ge \frac{{\left( {1 - \omega } \right)}}{\omega }k. $$

(8)

Note that for (8) to hold, a consumer simply takes the insurance premium and the set of coinsurance rates as given. Therefore, the capacity of a consumer to correctly calculate the premium is not required. From the individual consumer’s point of view, as long as (8) holds, it pays to visit a general practitioner first before visiting a specialist. For (8) to be satisfied, the probability of the severe illness condition has to be sufficiently small for any k. This makes intuitive sense because major illnesses do not occur as frequently as minor ones. There are many combinations of ϕ _H and ϕ _L that satisfy (8). In particular, condition (8), in fact, requires ϕ _L L < ϕ _H P _H . For example, no insurance (i.e., ϕ _H  = ϕ _L  = 1) is a candidate. However, ϕ _L  = 0 is a welfare-maximizing candidate if ϕ _H P _H  > 0. Then, (8) becomes

$$ u\left( {w - \alpha^{s} } \right) - u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right) \ge \frac{1 - \omega }{\omega }k $$

(9)

and (S2) is supported. Now, note that the actuarially fair insurance premium is α ^s = ρ[ω(1 − ϕ _L )P _L + (1 − ω)(1 − ϕ _H )P _H ]. When condition (9) is binding, a unique ϕ _H is defined. Therefore, one can conclude that:

Proposition 1

Given ϕ _L  = 0, ω is sufficiently small for a finite k such that (9) is binding at ϕ ^s _H  ∈ (0,1], the mixed-provider model is supported at any ϕ _H  ≥ ϕ ^s _H if ϕ ^s _H  ≤ 1.

Proof

Define D ^s(ϕ _L  = 0,ϕ _H ) = u(w − α ^s)−u(w − α ^s−ϕ _H P _H ) as a function of coinsurance rates. First, note that D ^s(0,0) = 0 and $\frac{{\partial \alpha^{s} }}{{\partial \phi_{H} }} = - \rho (1 - \omega )P_{H} .$ By differentiating D ^s(0,ϕ _H ) with respect to ϕ _H yields

$$ \frac{{\partial D^{s} }}{{\partial \phi_{H} }} = \left\{ {u^{\prime}\left( {w - \alpha^{s} } \right)\rho \left( {1 - \omega } \right) + u^{\prime}\left( {w - \alpha - \phi_{H} P_{H} } \right)\left[ {1 - \rho \left( {1 - \omega } \right)} \right]} \right\}P_{H} > 0. $$

Hence, if (9) is binding, then there is a unique ϕ ^s _H ∈ (0,1] such that $D^{s} \left( {0,\phi_{H}^{s} } \right) = \frac{{\left( {1 - \omega } \right)}}{\omega }k$ and for all ϕ _H  > ϕ ^s _H such that $D^{s} \left( {0,\phi_{H} } \right) > \frac{{\left( {1 - \omega } \right)}}{\omega }k$ if ϕ ^s _H  ≤ 1.

Proposition 1 suggests that the higher the coinsurance rate for heavy treatments, the higher the incentives for doctor searching. However, a higher coinsurance rate for the heavy treatment leads to less pooling and a higher out-of-pocket payment for health care by consumers. Hence, if the third party aims to maximize the welfare of the insured consumer, then the insurance contract should take the form I ^s = {α ^s, 0, ϕ ^s _H }, since full insurance is always the first best offer from the consumer’s point of view. Also, it can be shown that the higher the value of H and e, the lower the value of ϕ ^s _H needed for encouraging beneficial searching.

The central message of Proposition 1 is that when designing an insurance policy, the third party must create incentives for consumers to visit general practitioners first. This idea is in contrast with the common belief that general medicine shall not be covered by insurance to avoid the problem of moral hazard. It is worth noting that the abuse of light treatment does not occur in this model. The reasons are, first, treatments are bundles such that the quantity of treatment demanded and supplied does not change with the out-of-pocket payment; second, receiving treatment does not yield utility; third, there are no false-positive cases. Hence, consumers will not visit a provider for the sake of its consumption value or visit a provider wrongly. It is also necessary to point out that ϕ _L  = 0 is not an essential condition for Proposition 1 to hold. In fact, for any ϕ _L  > 0, there will be a corresponding ϕ ^s _H (ϕ _L ) such that if ϕ _H  > ϕ ^s _H (ϕ _L ) then beneficial searching will be the result. ϕ _L  = 0 is a plausible candidate because by increasing ϕ _L towards one, it increases the out-of-pocket expenditures when a mild illness condition occurs. However, the benefit of lowering the insurance premium by rising ϕ _L is spread out between mild and severe illness conditions, and thus it worsens the expected welfare of a consumer.

This mixed-provider model, in fact, is a voluntary gatekeeper system without interventions on the industrial organization of the provision market. If specialization is costless for all providers in the market, then market force will guide segregation according to the underlying preferences of the service providers. Consumers will be evenly shared among providers. Providers earn zero economic profit. To reach this result, the flow of information regarding prices among providers and consumers is assumed to be perfect. This market orientation is similar to the segregation of service providers into diagnostic shops (i.e., light treatment shops) and heavy treatment shops for illnesses of severe conditions. However, such segregation must be sustained by appropriate cost-sharing parameters to induce the supporting consumer purchasing behavior. The discussion below examines how full insurance coverage may once and for all eliminate the light treatment shops.

Specialist-dominated model

The following sub-sections consider two types of pooling equilibria. The first pooling equilibrium refers to a specialist-dominated model in which all providers become specialists and provide only heavy treatment. The second pooling equilibrium refers to situations in which all providers become specialists who provide both treatments.

The specialist-dominated model has the following configurations:

• (P1) Providers choose to become specialists, providing both or only heavy treatments. However, they honestly recommend treatment only if heavy treatment is needed; otherwise, they induce demand. Hence, y = 1 if both heavy and light treatments are needed.

• (P2) Consumers visit service providers only once and accept treatment recommendation (i.e., x = 1 on the first visit to a specialist who only recommends heavy treatments).

• (P3) The third party offers a welfare-maximizing insurance policy, I ^p = {α ^p, ϕ _L  = 0, ϕ _H  = 0}, where the subscript p stands for pooling, supporting (P1) and (P2).

Clearly, under the specialization configuration, all physicians becoming general practitioners cannot be an equilibrium configuration because major illnesses are untreated. The specialist-dominated equilibrium, as laid down above, in which all providers become specialists, is the only possible pooling equilibrium. Now, it is necessary to examine whether a provider can benefit by deviation from strategy (P1), given the consumer’s strategy (P2). A deviation from (P1) by the provider implies that she may provide light treatment honestly. However, given (P2), a deviation from (P1) is not optimal due to lower economic profit made by the deviant provider. Now, given (P1), there are no incentives for consumers to deviate from (P2), because it would only increase the consumer’s disutility from illness by additional rounds of doctor visits. Therefore, (P1) and (P2) are pure Nash equilibrium strategies.

Next, it is necessary to specify the insurance contract that supports (P2), and thus (P1). To eliminate a consumer’s incentives to pay the first visit to general practitioners, condition (8) must be violated for any price menu (P _L , P _H ). Therefore, the utility obtained by receiving light treatment after the payment of out-of-pocket cost of care, ϕ _L P _L , must be larger or equal to that after the payment of ϕ _H P _H for heavy treatments, such that $u\left( {w - \alpha^{s} - \phi_{L} P_{L} } \right) - u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right) < \frac{{\left( {1 - \omega } \right)}}{\omega }k.$ If the condition (P1) holds, then the reimbursement for light treatment is never executed. Hence, ϕ _L can take any arbitrary value. Full insurance ϕ _L  = ϕ _H  = 0 is thus a welfare-maximizing choice since it would be most preferred by insured consumers. Given (E3) above, the third party only reimburses at marginal cost, or if there is price competition, then the market price of heavy treatment will be driven to H + e. The premium under this specialist-dominated model is, therefore, α ^p = ρ(H + e). However, such equilibrium is an economically inefficient one because stinting occurs while service providers pocket a positive economic gain, which equals ωe.

Proposition 2

In the absence of service regulations and costs for specialization in professions, full insurance coverage induces demand-side moral hazard, over-provision of specialist treatment (i.e., supply-side demand inducement) and specialist-dominated primary care market.

Benefits shift between consumers and providers when moving from the mixed-provider model to the specialist-dominated equilibrium. In the mixed-provider model, only needed services are recommended and provided, and consumers have to bear out-of-pocket spending in times of illness of severe condition, despite paying in exchange a lower insurance premium than that under the specialist-dominated equilibrium. Shifting from the mixed-provider model to the specialist-dominated model, the consumers benefit from full insurance coverage, but at the cost of higher premiums charged by insurers, while providers earn information rents by stinting. The two equilibria, therefore, illustrate the cost and benefit of implementation of full insurance coverage, as well as the response of treatment decisions of service providers to a change of consumer behavior.

Certainly, in reality, we do not observe that all physicians become specialists because of the differential ability of physicians to become specialists. An exogenous institutional structure, such as a qualification accreditation mechanism, that regulates the composition of the doctor market will be necessary for the specialists and generalists to exist simultaneously in the doctor market if full insurance is provided by the insurer. In the absence of such a mechanism, the problems resulting from active supply-side demand inducement and passive demand-side moral hazard seem to be unavoidable.

In the absence of insurance, Dulleck and Kerschbamer [8] show that when under-treatment is ruled out institutionally but verification of a sophisticated treatment fails while consumers can visit more than one provider, service providers may overcharge and provide unnecessary treatment with some positive probability if there is not enough competition (Lemma 6); however, if there is enough competition, then the free market mechanism will induce efficient treatment (Lemma 7). For the latter, a similar result is shown by Wolinsky [26] with additional restriction on beliefs. Both assume sufficiently low search costs. The insurance-dependent equilibria presented above show that insurance coverage may improve or worsen the market outcomes (under the same set of assumptions: fulfillments of homogeneity and liability but violation of commitment and verification). In particular, efficient treatment can be induced and overcharging can be eliminated by appropriate cost-sharing and reimbursement policies, condition (E3) above, regardless of the level of competition among providers. However, supply-side inducement and demand-side moral hazard inevitably remain under full insurance in the absence of additional policy measures.

Salary payment model

Based on the above findings, contracting or interventions are, therefore, needed if the third party wishes to ensure economically efficient and free accessible health care. An important option is to require the third party to hire service providers directly by paying them a fee-for-service and a fixed salary, e, independent of the services provided to compensate for the effort exerted. Equivalently, the third party will bargain with the providers over a reimbursement price list that aligns the incentives of providers for honest provision of both types of treatment by paying information rents. Such a financial arrangement, however, induces providers to render light treatment because they can then pocket the salary by saving the effort. Under-treatment, however, would not occur in this setting. It is because health outcomes are observable to consumers and the third party, and hence service providers would be subject to a disciplinary penalty if intentional under-treatment is reported. There are no incentives for providers to practice self-defensive medicine by inducing demand for heavy treatment because there is no issue of diagnostic inaccuracy. Hence, this payment contract arrangement is sufficient to induce appropriate care. Providers will participate because this financial arrangement yields an economic profit for them, equivalent to that obtained under a pooling equilibrium, equal to ωe.

The new prepayment for health care charged becomes α ^c = ρ[ω(L + e) + (1−ω)(H + e)] where the subscript c stands for the contract, which lies between the ones under the specialist-dominated and the mixed-provider model. The salary payment model has the following configurations:

• (R1) Providers becomes specialists and provide both types of treatment honestly.

• (R2) Consumers visit providers and accept treatment recommendations (i.e., x = 1 on the first visit to a specialist who either recommends light or heavy treatments).

• (R3) The third party charges the consumers a prepayment for health care equal to the expected payout. She then pays a fixed salary e independent of treatment rendered and pays the cost of light treatment at L and heavy treatment at H.

Proposition 3

A salary payment contract, offered by the third party, which pays specialists a fixed salary, e, independent of the services provided, will alleviate both the supply-side demand inducement and demand-side moral hazard problems, resulting in full insurance and efficient provision of treatments.

In essence, this configuration is also a stylized form of managed care that integrates service provision and insurance. In this configuration, specialists have no incentives to induce demand, and as a result, consumers only search once. In comparison with the mixed-provider model, this salary payment model achieves full insurance and avoids additional disutility from illness at the cost of information rents payments to specialists. Providers do not have incentives to become general practitioners under the salary payment model. It is because the price charged by a general practitioner for light treatment must be L, which is lower than L + e, which would have been obtained by a specialist employed by the third party. Under the salary payment model, all service providers will become specialists.

Referral model

This sub-section identifies a mechanism by which the welfare outcome of the equilibrium can be improved, while saving the payments of information rents to providers. This can be achieved by dedicating purchasing power to the third party, letting her engage in direct contractual agreements with the insured consumers instead of the providers. In particular, if the third party effectively accepts or rejects a treatment recommendation on behalf of consumers, then welfare can be improved. This is called is a referral model such that the resulting premium becomes α ^I = ρ[ωL + (1 − ω)(H + e)], where the subscript I stands for involuntary, which has the following configurations:

• (I1) Providers become specialists and provide both types of treatments, pricing at marginal cost P _L  = L and P _H  = H + e. They will honestly recommend needed treatment without demand inducement (i.e., y = 0 if light treatment are needed and y = 1 if heavy treatment are needed).

• (I2) Consumers visit service providers for recommendations for treatment and inform the third party of the recommendations by providers for acceptance decisions.

• (I3) The third party offers full insurance and accepts a treatment recommendation only if (a) the price charged for the recommended treatment is the lowest in the market; (b) if light treatment is recommended on the first visit, then the third party accepts treatment, x = 1. If heavy treatment is recommended on the first visit, then the third party will require consumers to visit a second provider, x = 0. Treatment recommendations will be accepted on the second visit, x = 1, regardless of the recommendation.

In this configuration, providers have no incentives to charge a price higher than the marginal cost of treatment due to guaranteed rejection by the third party. Specialists have no incentives to induce demand by recommending heavy treatment if light treatment is needed on the first visit of a consumer, since in any case the first provider makes zero economic profit. The second provider will recommend heavy treatments because they are needed. As long as providers do not collude, this configuration achieves efficiency.

Proposition 4

By surrendering the purchasing decisions of consumers to the third party, full insurance can be offered without payment of information rents to the providers to induce appropriate recommendation of care, and consumers visit providers at most twice.

It is important to caution against the limitation of this proposed equilibrium. This equilibrium essentially follows a second-opinion rule, more or less like real-time utilization reviews. Whenever there is a heavy treatment recommendation, the third party will automatically refuse the recommendation on behalf of consumers and follow the treatment recommendation of the second opinion. An important qualification for this equilibrium to work properly is the accuracy of diagnosis, especially on the first round of consultation. If there is an error made in the first round of consultation that mistakes an illness of mild condition as a severe condition, then the second provider can induce demand. Even without diagnostic error, there is also an opportunity for collusion. The first provider can always recommend heavy treatment when the illness contracted by the patient is in fact mild. Then, the observable heavy treatment is rendered during the second consultation; however, then the first and second providers can together pocket and share the rents from avoidance of the non-observable effort. Monitoring of and penalizing for collusion involve economic resources and hence should be taken into consideration.

This referral model invades consumer sovereignty because consumers have to surrender purchasing decisions to the third party. Complete surrender of purchasing decisions may sound uncommon, but the referral model, as the name suggests, can be implemented essentially as follows: (1) Heavy treatment can be provided only if a referral letter is presented; (2) the issuer of the referral letter must not be the provider of heavy treatments; (3) light treatment is reimbursed without a referral letter, but heavy treatment is reimbursed only if a referral letter is presented; (4) the insurer reimburses at the lowest prices in the market. Under this system, the third party actively influences the searching decision of consumers in the direction desired by the third party. In particular, consumers will reject heavy treatment in the first round of a visit because heavy treatment will not be reimbursed without a referral letter. However, consumers will accept heavy treatment on their second visit because treatment will be reimbursed based on the referral letter issued by the first provider. Correspondingly, providers will provide light treatment if a mild illness condition is found, issue a referral letter for heavy treatment if a severe illness condition is found and provide a heavy treatment if a referral letter is presented by a patient. The referral model, thus, achieves the efficient outcome (Lemma 7) in the analysis by Dulleck and Kerschbamer [8], as discussed in the “specialist-dominated model” above. Again, there is no assumption needed for the number of providers in the market for Proposition 4 to hold, as long as there is price competition or the third party only reimburse the lowest priced providers (i.e., condition (E3) holds).

Ranking

This section ranks the mixed-provider, specialist-dominated, salary payment and referral models from the consumers’ perspective. Before proceeding, note that except for the mixed-provider model, the three other models yield a pooling equilibrium in which providers choose to become specialists. Only under the mixed-provider model, doctors separate themselves into specialists and general practitioners. In terms of the delivery of treatments, providers render heavy treatment to patients regardless of their illness condition only under the specialist-dominated model, thus incurring the highest health-care expenditures. However, in all other configurations, there is service delivery according to need. The mixed-provider and the referral model would incur the lowest health-care costs, but patients may search twice. Meanwhile, the salary payment model incurs the median health-care cost due to a payment of information rents, but patients would have to visit only once. The ranking of these models would then depend on the cost of treatments, disutility from illness and the information rents. Let us outline the welfare of a consumer under the different models:

• (i) Mixed-provider model: $V^{s} = u\left( {w-\alpha^{s} } \right)-\rho \left[ {\frac{{(1 - \omega )^{2} }}{\omega } + (2 - \omega )} \right]k$ where α ^s = ρ[ωL + (1 − ω)(1 − ϕ ^s _H )(H + e)]

• (ii) Specialist-dominated model: V ^p = u(w − α ^p) − ρk where α ^p = ρ(H + e)

• (iii) Salary payment model: V ^c = u(w − α ^c) − ρk where α ^c = ρ[ω(L + e) + (1 − ω)(H + e)]

• (iv) Referral model: V ^I = u(w − α ^I) − ρ(2 − ω)k where α ^I = ρ[ωL + (1 − ω)(H + e)]

To compare the models, note that the key variation comes from the differences in premium and the expected number of visits, and thus the magnitude of disutility from illness occurs due to doctor searching. The following proposition reports the conditional pair-wise ordering of the four models.

Proposition 5

Based on the definitions of consumer’s payoffs (i) to (iv), the following results are derived. Assume ω ∈ (0,1).

• (a) or any non-negative H, L, and e, V ^I > V ^s and V ^c > V ^p.

• (b) For any non-negative H and L, if there exists a threshold effort level e ^* such that V ^I − V ^c = 0, then for any e > e*, V ^I > V ^c, otherwise for any e < e*, V ^c > V ^I.

• (c) For any non-negative H and e, if there exists a threshold L* such that V − V ^p = 0, then for any L < L*, V ^s > V ^p, otherwise for any L > L*, V ^p > V ^s.

• (d) For any non-negative L and e, if there exists a threshold effort level H ^* such that V ^I − V ^p = 0, then for any H > H*, V ^I > V ^p, otherwise for any H < H*, V ^p > V ^I.

• (e) For any H such that ωe < ϕ ^s _H (H + e) and any non-negative L and e, V ^c > V ^s.

Otherwise, if ω = 0, then the specialist-dominated equilibrium is equivalent to the salary paid model, and is the most preferred. If ω = 1, then the referral equilibrium is equivalent to the mixed-provider model and is the most preferred.

The proof of the proposition is provided in Appendix. The comparisons among the payoffs of consumers in pairs help rank the preference ordering of different health insurance models. Part (a) suggests that the referral model welfare dominates the mixed-provider model while consumers prefer the salary payment model to the specialist-dominated model. Part (b) suggests that the ranking between the referral model and the salary payment model depends on the effort cost, and hence the information rents; part (c) suggests that the ranking between the mixed-provider and the specialist-dominated models depend on the cost of light treatments; part (d) suggests that the ranking between the referral and the specialist-dominated model depends on the cost of heavy treatments. Part (e) suggests that if the information rents to be paid are less than the co-payment for heavy treatments, then the salary payment model yields higher consumer welfare than that under a mixed-provider model. Figure 1 shows the four areas that compare in pairs the consumer welfare under these models.

Fig. 1

Comparison of consumer payoffs under different health insurance models

Then, Proposition 6 straightforwardly follows.

Proposition 6

Assume ω ∈ (0,1), H*, L* and e* exist. Then, by Proposition 5,

• (a) If L > L*, e > e*, and $H > \left( {\frac{\omega }{{\phi_{H}^{s} }} - 1} \right)e$, then V ^I > V ^c > V ^s > V ^p .

• (b) If L < L*, e < e* and H > H*, then V ^c > V ^I > V ^s > V ^p.

• (c) If L > L*, e > e* and H > H*, then V ^I > V ^c > V ^p > V ^s.

• (d) If L < L*, e < e* and H > H*, then V ^c > V ^I > V ^s > V ^p.

Therefore, according to the ranking above, V ^s and V ^p are dominated by V ^I and V ^c.

Proposition 6 suggests that the ranking of the four models depends mainly on the costs of tangible treatments, the disutility from illness, the information rents and the co-payment for heavy treatments. In all the cases above, consumers prefer some forms of contracting either with consumers or providers. Both contractual and demand-side cost sharing arrangements almost always dominate a full insurance arrangement that results in supply-side demand inducement. Parts (a) and (c) show that if the information rents to be earned by providers and the cost of light treatment are not large, a salary payment contract is the preferred arrangement to pay the providers. As such, the providers can either provide light or heavy treatment; however, by providing light treatments, they earn a compensation for honest recommendations. However, parts (b) and (d) in Proposition 6 show that if the information rents as well as the cost of light treatment are sufficiently high, then surrendering the purchasing decision to the third party will be preferred. The results above echo the finding that managed care techniques dominate conventional plans that rely on control by cost sharing in Blomqvist and Leger [2]. In the model above, cost sharing as a mechanism to promote the gatekeeper system is strictly dominated by the insurance with elements of managed care.

Note that in any case, the referral model is ranked highly by consumers. This seems to contradict the common understanding that consumers seem to prefer direct access to specialist care. Without invoking the argument for the protection of the patient’s rights and freedom, if the probability of contracting an illness of severe condition is very high such that ω → 0, then direct access to specialist care will be preferred by consumers since over-treatments are unlikely, making the specialist-dominated equilibrium the most preferred option. However, when ω > 0, then direct access to specialist care comes with the cost of paying a higher premium by consumers, which the consumers correctly factor in their calculation because of the assumption of homogeneity. Hence, unlike realistic situations in which there are heterogenous consumers with different illness probabilities such that some individuals may prefer direct access to specialist care because of private knowledge of an illness condition, the situation described in the paper leads to the desire for the referral model. Certainly, lost freedom for direct access to specialist care may incur a cost that has not been taken into account. On the supply side, a referral model may offend service providers. This is because under such a mechanism, the recommendation of the first service provider is treated as if it were discredited by the third party. Also, the monitoring cost against collusion among providers could be high. Therefore, the salary payment model would likely to be the most preferred model when these considerations are taken into account.

Conclusion

This paper shows the linkage between the industrial organization of the provision market and the features in the health insurance offered by the third party. Emphasis is placed on the economics of primary care in the presence of information asymmetry by de-emphasizing the trust between providers and patients. As Rothschild and Stiglitz [22] point out, the presence of asymmetric information does not permit full insurance without resulting in inefficiency. This paper shows that full insurance offered in the absence of regulation in the provision market will result in the over-provision of more expensive medical treatment and demand-side moral hazard problems. Under full insurance, consumers do not have incentives to reject more expensive inefficient treatment, and this thus encourages supply-side demand inducement. To an extreme, if it is costless for providers to specialize, then specialized expensive treatment will be provided exclusively regardless of the underlying health-care need.

Although differential treatment costs and price competition are sufficient to encourage consumer behavior that implements a voluntary gatekeeper system, treatment costs, do not represent the out-of-pocket payment by consumers because of the popularity of health insurance. Demand-side cost sharing can then be used to implement a voluntary gatekeeper system and to induce providers to endogenously separate themselves into general practitioners and specialists. In particular, under Nash equilibrium, market forces will guide the segregation of the providers into general practitioners who provide low-cost treatment and specialists who provide high-cost treatment exclusively. Under such a voluntary gatekeeper system, the relative out-of-pocket payment for utilizing specialized treatment by consumers is artificially raised higher than that for less expensive general treatments. As a consequence, consumers will obtain a recommendation from general practitioners first before visiting specialists, which occurs only if necessary. Hence, demand-side cost sharing alone can deal with the supply-side inducement and demand-side moral hazard problems as well as induce efficient treatment by partial insurance.

Contracting between providers and the third party payer is common among HMOs in Switzerland and in the US, paying providers a mixture of capitation, salary and fee-for services possibly with supply-side cost sharing as well as other cost-containment measures. In Europe, most providers enter into contractual relationships with sickness funds, national health insurance agencies or governments. Managed care techniques are often experimented. Although capitation is not analyzed in this paper, it is a popular and powerful payment method for controlling cost that shifts financial risk from consumers to providers, but encourages under-treatment. In the context of the model described above, if under-treatment is observable and can be penalized, then a capitation model will lead to a pooling equilibrium with efficient provision because providers are risk neutral. However, a thorough analysis of the capitation model requires relaxation of risk neutrality of providers and liability of under-treatment. Therefore, this should be left for future research.

This paper shows that contractual arrangement welfare dominates the stand-alone demand-side cost sharing as a way to tackle both the demand inducement and the ex-ante moral hazard problems in primary health-care provision. By hiring and allowing providers to provide a variety of services, a salary payment scheme or a price list for treatment must be designed by the third party to align the incentives for provision of all types of services, such that there is no incentive to over-provide and induce demand. If health outcomes are observable after treatment and if under treatment by providers can be penalized, then efficient rationing of care can be induced. In the salary payment model, all providers become specialists and provide both high- and low-cost treatments; meanwhile consumers only need to visit providers once and receive efficient treatment. The payment of information rents will then be shared among contributors. The third party consequently charges a premium for health care that is equal to the expected payout without imposing any out-of-pocket spending at the point of services (i.e., full insurance is provided). Providers earn information rents for provision of less expensive treatments.

To reserve full insurance and yet avoid the payment of information rents, the purchasing decision of the consumers could be surrendered to (or be influenced by) the third party by following a rule that requires consumers to reject heavy treatment on their first visit and accept heavy treatment only on their second visit. By doing so, it creates incentives for searching for an efficient treatment by the consumers. However, this arrangement intrudes into consumer freedom for direct access to specialist care and may inadvertently harm service providers’ professional dignity if their targets are beyond profit maximization. Furthermore, as Bernheim, Peleg and Whiston [2] and Moreno and Wooders [17] show, a coalition-proofness Nash equilibrium does not always exist. Hence, coalition remains a general problem among all the Nash equilibria discussed in the paper. Hence, the viability of such arrangement requires costly monitoring and measures against collusion among service providers.

Whether a salary payment model or referral model is preferred will depend on the sizes of information rents and their implementation costs. If the information cost is moderate and the implementation costs of the two models are comparable, then the salary payment model is preferred. Otherwise, consumers may prefer the referral model, depending on the additional costs discussed above. Salary payments and referral often manifest themselves as managed care techniques that involve integration of downstream service providers and regulation of consumer’s access to care by the third party. Although these techniques may invite a backlash, they generally dominate the demand-side cost-sharing approach for dealing with the problems of supply-side demand inducement and demand-side moral hazard problems. Hence, as a conclusion, contracting and intervention in the health-care industry are preferred to complete decentralization of delivery in the presence of information asymmetry.

Appendix: Proof of Proposition 5

To prove Part (a), note that V ^I − V ^s = u(w − α ^I) − u(w − α ^s) + [(ρ(1 − ω)²)/ω]k > 0 since α ^I < α ^s. On the other hand, V ^c − V ^p = u(w − α ^c) − u(w − α ^p) > 0 since α ^p − α ^c = ρH − ρ[ωL + (1 − ω)H] > 0.

To show Part (b), note that V ^I − V ^c = u(w − α ^I) − u(w − α ^I − ρωe) − (1 − ω)k. Let us define Δ(e) = u(w − α ^I) − u(w − α ^I − ρωe). Note that Δ(0) = 0 and

$$ \frac{\partial \Updelta (e)}{\partial e} = \left[ {u^{\prime}\left( {w - \alpha^{I} - \rho \omega e} \right) - u^{\prime}\left( {w - \alpha^{I} } \right)} \right]\left( {\frac{{\partial \alpha^{I} }}{\partial e}} \right) + u^{\prime}\left( {w - \alpha^{I} - \rho \omega e} \right)\rho \omega > 0. $$

Then, if V ^I − V ^c = 0 at e = e*, then Part (b) follows.

To show Part (c), we first show the sign of $\frac{{\partial \alpha^{s} }}{\partial L}.$ Note that $\frac{{\partial \alpha^{s} }}{\partial L} = \rho \left[ {\omega - \left( {1 - \omega } \right)\frac{{\partial \phi_{H}^{s} }}{\partial L}} \right].$ Define $\psi \left( {L,\phi_{H}^{s} } \right) = u\left( {w - \alpha^{s} } \right) - u\left( {w - \alpha^{s} - \phi_{H} P_{H} } \right) - \frac{{\left( {1 - \omega } \right)}}{\omega }k = 0.$ It follows that $\frac{{\partial \phi_{H}^{s} }}{\partial L} = - \frac{{\frac{\partial \psi }{\partial L}}}{{\frac{\partial \psi }{{\partial \phi_{H}^{s} }}}} < 0$ since $\frac{\partial \psi }{\partial L} = \rho \omega \left( {u^{\prime}\left( {w - \alpha^{s} - \phi_{H}^{s} P_{H} } \right) - u^{\prime}\left( {w - \alpha^{s} } \right)} \right) > 0$ and $\frac{\partial \psi }{{\partial \phi_{H}^{s} }} = u^{\prime}\left( {w - \alpha^{s} } \right)\rho \left( {1 - \omega } \right)P_{H} + u^{\prime}\left( {w - \alpha^{s} - \phi_{H}^{s} P_{H} } \right)\left[ {1 - \rho \left( {1 - \omega } \right)} \right]P_{H} > 0.$

Then, note that $V^{s} -V^{p} = u\left( {w-\alpha^{s} } \right)-u\left( {w-\alpha^{p} } \right)-\left[ {\frac{\rho (1 - \omega )}{\omega } + 1} \right](1 - \omega )k.$ Let G(L) = u(w − s) − u(w − α ^p), which satisfies $\frac{\partial G(L)}{\partial L} = -u^{\prime}\left( {w-\alpha^{s} } \right)\frac{{\partial \alpha^{s} }}{\partial L} < 0.$ Hence, if $G\left( 0 \right) > \rho \left[ {\frac{{\left( {1 - \omega } \right)}}{\omega } + 1} \right]\left( {1 - \omega } \right)k,$ then there exists a L* such that V ^s − V ^p = 0. By $\frac{\partial G(L)}{\partial L} < 0,$ then part (c) follows.

To show Part (d), note that V ^I − V ^p = u(w − α ^I) − u(w − α ^p) − (1–ω)k. Let F(H) = u(w − α ^I) − u(w − α ^p). F(0) < 0 and $\frac{\partial F\left( H \right)}{\partial H} = \left[ {u^{\prime}\left( {w-\alpha^{p} } \right)-u^{\prime}\left( {w-\alpha^{s} } \right)\left( {1 - \omega } \right)\rho } \right] > 0.$ Then part (d) follows if V ^I − V ^p = 0 at H*.

To show Part (e), note that V ^c − V ^s = u(w − α ^c) − u(w − α ^s) + [(ρ(1 − ω)²)/ω + (1 − ω)]k > 0 if α ^s > α ^c, which is satisfied if α ^s − α ^c = ρ[ωL + (1 − ω)(1 − ϕ ^s _H )(H + e)] − ρ(H + e) > 0. Hence, it requires ωe < ϕ ^s _H (H + e). □