Tiered and Value-based Health Care Networks

Tiered and Value-based Health Care Networks Abstract This paper studies the design of health care networks. Information systems for consumers, payment mechanisms for providers, and copayments for consumers are simultaneously considered. The optimal design aims to implement e¢ cient provider qualities and cost-reduction e¤orts, and e¢ cient allocation of consumers across di¤erent providers. For information system, we consider Tiered and Valued-based health care networks. In a Tiered Network, a provider is assigned a quality-cost designation, say, Excellent, if its quality and cost e¤ort are above some thresholds; otherwise, it is assigned a Standard designation. In a Valued-based Network, a quality-cost index is constructed for a provider. For payment mechanisms, we consider cost reimbursement and prospective payments. Consumer copayments may be based on a provider’s quality and cost. We show that the …rst best can be implemented in Tiered and Value-based Networks under either cost reimbursement or prospective payment. We compare the implementation costs across di¤erent designs. 1 Introduction Health care markets su¤er from a number of market failures. First, there is moral hazard on the demand side. Due to health insurance, consumers pay subsidized prices, and tend to overvalue service bene…t and undervalue cost e¢ ciency. Second, and arguably more serious, providers’service quality and cost e¢ ciency are di¢ cult to contract upon, and may be unobserved by consumers. Information design in health care networks is the latest innovation to address these market failures. In this paper we present a model of information-based networks, and study how they can complement supply-side and demand-side …nancial incentives to overcome market failures. We consider two information systems. In a Tiered Network, a health plan uses providers’quality and cost information to sort providers into di¤erent tiers, which serve as quality-cost indicators. For example, after a provider’s quality and cost information has been assessed, it is designated as either Standard or Excellent. In a Value-based Network, a health plan uses providers’quality and cost information to construct value indexes, which inform consumers about qualities. For example, a provider’s value index of 80 may be the simple average of its quality score of 90 and e¢ ciency score of 70. In both networks, a health plan can also use quality and cost information to set consumer cost shares. These networks are now common.1 For example, in 2011, Blue Cross Blue Shield of Massachusetts introduced Hospital Choice Cost Sharing networks and Aetna began o¤ering Choose and Save networks in 23 states.2 In 2013, 23% of employers in the United States included tiered networks in their health plans (Kaiser, 2013). For a Tiered Network or a Valued-based Network, we study how the two most common payment mechanisms, cost reimbursement and prospective payment, can be used for the implementation of providers’ service quality and cost-reduction e¤orts. We also let the competing providers be heterogenous; some are more technologically e¢ cient than others. This asymmetry is a necessary element in a model for studying mechanisms that rank providers. We derive some principles of good network design. First, tiers and value indexes are e¤ective for aligning 1 See, for example, Robinson (2003). 2 http://www.bluecrossma.com/plan-education/medical/hccs/. 1 providers’quality and cost incentives, precisely because they bundle quality and cost information. Indeed, we construct optimal tiers and value indexes. Second, consumers need not bear very high costs for optimal service utilization. In fact, we derive the lowest cost shares for e¢ ciency, which generally depend on the quality and cost di¤erences among providers. Finally, our results on network design change the conventional wisdom about the di¤erence in incentive properties between cost reimbursement and prospective payment. We actually show that either cost reimbursement or prospective payment can implement e¢ cient quality and cost e¤orts in a network. In our model, two providers produce horizontally and vertically di¤erentiated health services for consumers covered by a health plan. A provider’s unit cost increases in service quality, but can be reduced by an investment. Both quality and cost-reduction investments are costly, but providers di¤er in investment e¢ ciency. One provider’s investment cost is a fraction of the other’s. Quality and cost-reduction e¤orts are not contractible. Each of a set of covered consumers picks one of the two providers for service. However, consumers cannot directly observe providers’qualities (or costs). They have to obtain such information from the health plan. We postulate that the health plan can obtain information about providers’ qualities and cost e¤orts, and that it chooses how to report the information to consumers. Although the health plan could have disclosed all available information to consumers, we show that partial disclosure of quality and cost information can motivate providers to invest in quality and cost e¤orts.3 We identify disclosure policy with network design. In a Tiered Network, the health plan assigns a provider to a tier if its quality and cost e¤ort are above certain thresholds. The network therefore chooses thresholds and discloses whether a provider’s quality and cost e¤ort are above those thresholds. The health plan also sets consumers’copayment for using a provider according to that provider’s tier. In a Value-based Network, the health plan constructs a weighted average of a provider’s quality and cost e¤ort, and reports that index to consumers. The network therefore chooses weights on quality and cost and discloses summary statistics of a provider’s services. The health plan can also set consumers’copayment for using a provider according 3 The usual argument for partial disclosure is that limited expertise and cognitive capacities make full disclosure impractical. Although we are sympathetic to this view, we have chosen to ignore “bounded rationality”in this paper. 2 to how much the provider costs the plan. In a Tiered Network, a strategic provider will attain any tier by choosing quality and cost e¤orts just satisfying the thresholds. Changing the de…nition of a tier changes the provider’s quality and cost-e¤ort choices. In a Value-based Network, a strategic provider will attain a certain level of the index to maximize pro…t, taking into account consumers’inference about quality from the index. The health plan must anticipate this provider reaction when constructing the index. The health plan also sets copayments to induce e¢ cient service utilization. What matters is consumers’ incremental cost between providers. The e¢ cient consumer incremental cost should let them internalize providers’cost di¤erence. In a Tiered Network, the health plan can set the Excellent-tier copayment to zero, and the Standard-tier copayment to the unit-cost di¤erence between providers in the two tiers. The same principle applies to a Value-based Network. The health plan may use either cost reimbursement or perspective payment to implement quality and cost targets in Tiered and Value-based Networks. This is because both tiers and indexes can create complementary incentives for quality and cost e¤orts. However, implementation requires the health plan to carefully coordinate its provider payment and information systems. Moreover, because quality and cost incentives are di¤erent under prospective and cost-reimbursement systems, the costs for implementing the same quality and cost targets di¤er. We identify service cost structures which make one payment system less expensive than the other. Our network designs are information based. In 2010, about 44% of consumers have access to health-plan sponsored hospital quality reports (Christianson et al., 2010). Dranove and Sfekas (2008) …nd that quality reports do a¤ect consumer choices after consumer prior information is accounted for. Recently, Medicare and other sponsors also disclose providers’ cost-e¢ ciency information. However, researchers are skeptical about its e¤ect on consumer choice (Mehrotra et al. 2012). Our model shows that demand response to cost disclosure critically depends on how quality and cost information are conveyed. We propose that quality and cost information be bundled into value measures. The Massachusetts Group Insurance Commission is one of the …rst purchasers to adopt this approach. The Commission started o¤ering 3 tiered networks to more than 267,000 state employees and retirees in 2004 (Alteras and Silow-Carroll, 2007). In these tiered networks, health plans use quality and cost-e¢ ciency information to assign providers to one of Tiers 1, 2, and 3. For example, a primary care physician is assigned to Tier 1 only if its quality score is above B on a scale from A to C and its e¢ ciency score is above average.4 Consumers are informed about a provider’s tier only but not the underlying quality or e¢ ciency score. This approach is consistent with our Tiered Network. We also study value index as an alternative disclosure strategy. More important, we illustrate how tier and index construction, consumer cost sharing, and provider payment should be coordinated. To the best of our knowledge, this is the …rst attempt in the literature to do so.5 Innovations in consumer cost sharing are increasingly prominent. A number of recent papers study valuebased consumer cost sharing. Thomson et al. (2013) report that value-based cost sharing has been adopted in the United States and more than 10 European countries. Pauly and Blavin (2008) study the price theory of value-based cost sharing; their analysis is based on a price-taking service market where quality and cost e¢ ciency are …xed. They show that when consumers are fully informed, optimal cost-based and value-based copayments are identical. Valued-based copayment is lower only when consumers are imperfectly informed and underestimate service bene…t. They argue that information disclosure and value-based cost sharing are substitutes. By contrast, we study copayment policy in an oligopolistic service market where both quality and cost are endogenous. In our Tiered and Value-based Networks, information disclosure and copayments together guide consumers to make e¢ cient provider choices. We show that the optimal copayment for e¢ cient utilization of high-value services can be as low as zero. In our model consumers are rational. Baicker et al. (2012) study demand for health services when consumers su¤er from various behavioral biases. They also …nd that the optimal copayment for high-value services can be zero. Our results complement theirs. Moreover, we show that zero copayment is consistent with production e¢ ciency. The utilization and production e¢ ciency results distinguish our work from other papers that study 4 http://www.unicarestateplan.com/pdf/CPIPhysicianBrochureFY13.pdf 5 The three policy instruments have been studied extensively but separately. For recent surveys of the three strands of literature, see Dranove and Jin (2010), Léger (2008), and McGuire (2011). 4 oligopolistic provider competition. Beitia (2003) and Levaggi (2005) also consider a health plan contracting with two providers that produce horizontally di¤erentiated services. In their papers quality and cost e¢ ciency are not contractible either. However, they let consumers always observe quality and pay zero copayment. In both papers, …rst-best utilization is unattainable because the health plan lacks the copayment instrument to direct consumer choices. In their second-best solutions, qualities are distorted from the social optimal to minimize consumption ine¢ ciency. By contrast, we obtain conditions for e¢ ciency; this illustrates that information disclosure in network design, provider payment, and consumer copayment should be considered simultaneously.6 The paper is organized as follows. Section 2 sets up the model of a health plan incentivizing providers to service enrollees. We study Tiered and Valued-based Networks, respectively, in Sections 3 and 4. There, for each information-based network, we show the implementation of …rst-best qualities and cost e¤orts under cost reimbursement and prospective payment, and compare the health plan’s implementation costs under the two payment systems. In Section 5, we then compare the implementation costs of Tiered and Value-based Networks. Finally, Section 6 concludes. All proofs are in the Appendix. 2 The model We now set up a model of health care networks. A group of consumers will receive health services from one of two providers under a health plan. We discuss, in turn, these providers, the consumers, and the health plan structure. 2.1 Providers Two providers, A and B, supply health services to consumers in a health plan. Each provider chooses a service quality and a cost-reduction e¤ort. Let (qi ; ei ) denote the quality and cost e¤ort, both nonnegative, chosen by Provider i, i = A; B. Under quality q and cost e¤ort e, each provider’s unit cost of serving a consumer is given by the function C(q; e). The unit cost function is strictly increasing in quality and strictly 6 Ma (1994) shows that prospective payment can achieve …rst-best quality and cost e¢ ciency. Ma and Mak (2013) show that cost reimbursemnt together with information disclosure can implement the same outcome. Both papers consider a purchaser contracting with a single provider. 5 decreasing in e¤ort (Cq > 0 > Ce ), so higher quality means higher unit cost, but higher e¤ort can reduce it. We assume that the function C is strictly convex, and also that the marginal unit cost of quality, Cq (q; e), is nonincreasing in cost e¤ort (Cqe (q; e) 0). While the two providers share the same variable cost structure, their …xed costs are di¤erent. One provider is more e¢ cient than the other. A provider’s …xed cost is determined by the quality and cost e¤ort. If Provider A takes quality and cost e¤ort (qA ; eA ), its …xed cost is G(qA ) + H(eA ), where G and H are both strictly increasing and strictly convex functions. If Provider B takes quality and cost e¤ort (qB ; eB ), its …xed cost is [G(qB ) + H(eB )], with > 1. Thus, the parameter measures the ine¢ ciency of Provider B relative to Provider A. The ine¢ ciency may be due to higher management costs, or tighter local market conditions which drive up e¤ort and quality costs, etc. Although Provider B is less e¢ cient, it may still operate because some consumers may naturally prefer using its services, as we describe next. 2.2 Consumers The total mass of consumers covered by the health plan is normalized at 1. Each consumer is described by a valuation of health services, v, and a proclivity parameter x towards the two providers. Each consumer has the same valuation, so v is a strictly positive constant and identical for all consumers. We use the common Hotelling, horizontal product-di¤erentiation structure to describe consumers’ proclivity towards the two providers. Consumers are uniformly distributed on the [0; 1] interval. A nonnegative parameter measures the strength of consumers’ horizontal preferences. A consumer located at x incurs a mismatch disutility x when obtaining services from Provider A, and a mismatch disutility (1 services from Provider B. When x) when obtaining is large, each provider faces a less elastic demand. A consumer may have to pay a copayment to use a provider. We will expand on copayment determinations when we describe health plan networks in Subsection 2.4. Let sA and sB , both nonnegative, be the respective copayments when a consumer uses Provider A and Provider B. When health care qualities at Providers A and B are, respectively, qA and qB , a consumer at x obtains utilities vqA sA x and vqB sB (1 x) from these providers. We let each consumer receive services from one of the two providers, so the market is covered. 6 We assume that consumers have the same valuation on quality. Our interpretation is that we consider the subset of consumers within a health plan who view the quality of care of this service similarly. Within this set of consumers, the horizontal product di¤erentiation aspect of consumer preferences can be broadly interpreted. The Hotelling line can literally mean distance, and in health care, travel costs are important determinants of demands. Alternatively, di¤erent providers may have di¤erent practice styles, even di¤erent hours of operations, as well as other support services. Consumers may have diverse preferences towards these attributes, which are captured by the location and intensity parameters, x and . 2.3 First best The health plan writes contracts with providers and consumers. In the …rst best, providers’ qualities and e¤orts, and consumers’provider choices are contractible. An allocation speci…es each provider’s quality and cost-reduction e¤ort, as well as which provider should serve each consumer. The …rst-best allocation is one that maximizes aggregate consumer utilities less the production cost, which we call social welfare. It is obvious that the …rst best will assign consumers with small values of x to Provider A, and consumers with large values of x to Provider B: The health plan chooses these variables to maximize the sum of consumer utilities less the production cost. Let the health plan assign consumers with x remaining consumers to Provider B. Social welfare is W (qA ; qB ; eA ; eB ; x b) Z 0 x b [vqA x C(qA ; eA )]dx + Z 1 x b [G(qA ) + H(eA )] x b to Provider A, and the [vqB [G(qB ) + H(eB )]: (1 x) C(qB ; eB )]dx (1) The …rst term in the welfare expression (1) is the sum of consumers’ utilities less the variable costs for consumers obtaining services at Provider A; the second term is the corresponding value at Provider B. The last two terms in (1) are the total …xed costs of qualities and e¤orts. The following Lemma characterizes the …rst best. (All proofs are in the Appendix.) Lemma 1 In the …rst best, Provider A sets a higher quality, a higher cost e¤ ort, and serves more consumers than Provider B. That is, if qA , qB , eA , eB , and x b denote the …rst-best qualities and e¤ orts by Providers A and B and the consumer allocation across them, then qA > qB , eA > eB , and x b > 1=2. Furthermore, 7 they satisfy the following: (1 x b [v x b )[v (1 1 v(qA + 2 qB ) Cq (qA ; eA )] = G0 (qA ) Cq (qB ; eB )] = x b Ce (qA ; eA ) x b )Ce (qB ; eB ) [C(qA ; eA ) 2 C(qB ; eB )] G0 (qB ) = H 0 (eA ) = H 0 (eB ) = x b : (2) (3) (4) (5) (6) Because Provider A is more e¢ cient than Provider B (which has higher …xed costs ( > 1)), the …rst best prescribes that Provider A has higher quality, high e¤ort, and serves more consumers. Equations (2) to (6) have the usual interpretations. Raising a provider’s service quality increases consumer utilities, but also the provider’s unit cost and …xed cost of quality. Equations (2) and (3) balance these marginal e¤ects. Similarly, raising a provider’s cost e¤ort decreases unit cost, but increases …xed cost of e¤ort. Equations (4) and (5) balance these two opposing e¤ects. Equation (6) de…nes x b , the consumer who receives the same net social bene…t from either provider. 2.4 Health plan, payment, and information We …rst describe the provider payment contracts. Because of the complexity of health care services, contracts that specify how providers choose qualities and cost-reduction e¤orts are infeasible. In practice, health plans do use …nancial (and renewal) contracts that are based on how much health care services and cost providers have supplied. Accordingly, we let qualities and cost-reduction e¤orts be noncontractible, but the quantities and unit costs of services C(qi ; ei ), i = A; B, are ex post veri…able. We study the two most common forms of provider payment. Under prospective payment, the health plan pays pi to Provider i for a unit of service. Under cost reimbursement, the health plan pays any ex post unit cost C(qi ; ei ) plus a (positive) margin mi to Provider i for a unit of service. In this paper, we only consider these two forms of payments in the analytical models.7 7 Consumer cost heterogenity can be readily incorporated into this framework. See, for example, Ma (1994), Ma and Mak (2013). 8 Consumers’ choices of providers are also noncontractible. As mentioned above, the health plan may impose copayments. Consumers do not observe providers’ quality and cost-e¤ort choices; the health plan, however, does observe these choices. The health plan decides how to convey quality and cost-e¤ort information to consumers. In the literature, consumers are assumed to observe providers’ care quality. In our setup, this is equivalent to the health plan fully disclosing quality information to consumers. Here, we study a general information disclosure strategy. In a Tiered Network, the health plan uses the information on quality and cost e¤ort (qi ; ei ), i = A; B, gathered from the providers in order to construct tiers. We consider two tiers: Excellent and Standard. A provider belongs to the Excellent tier if its quality and cost e¤ort are above some given thresholds; similarly for the Standard tier. (We de…ne these thresholds formally in the Section 3.) After assessing providers’qualities and cost e¤orts, the health plan announces each provider’s tier. Consumers’copayments for services obtained from a provider may depend on that provider’s designated tier. For example, if Provider A is in the Excellent tier, then consumers may pay lower copayments when using it than if Provider A is in the Standard tier. In a Value-based Network, after observing a provider’s quality and cost e¤ort, the health plan constructs a value index equal to a linearly weighted sum of the quality and the cost e¤ort. (The value index will be formally de…ned in Section 4.) The health plan then discloses providers’quality indexes to consumers. The health plan also sets consumers’copayments for obtaining services from the two providers. 3 Tiered Network 3.1 Network structure and extensive form We …rst lay out how the health plan constructs the tiers and copayments. Figure 1 illustrates the network structure. We use two triples, (q Ex ; eEx ; sEx ) and (q St ; eSt ; sSt ), to de…ne health plans’tiers and copayment policies, where q, e, and s denote quality, cost e¤ort, and consumer copayment. The health plan assigns Provider i to the Excellent tier if and only if quality and cost e¤ort (qi ; ei ) are both higher than (q Ex ; eEx ): (qi ; ei ) (q Ex ; eEx ) (0; 0). This is the upper-right region in Figure 1. Next, suppose that Provider i 9 Figure 1: Tiers and copayments fails to qualify for the Excellent tier (because qi < q EX , e < eEX , or both). Then, Provider i is assigned to the Standard tier if and only if quality and cost e¤ort (qi ; ei ) are both higher than (q St ; eSt ): (qi ; ei ) (q St ; eSt ) (0; 0), where q St < q Ex and eSt < eEx . This is the shaded, L-shaped region in Figure 1. A consumer pays sEx 0 to obtain a unit of service from a provider in the Excellent tier, and sSt 0 from one in the Standard tier. If a provider fails to achieve any tier, it is excluded from the network. (We will assume that insured consumers will not obtain service from an excluded provider.) We study the following Tiered Network, extensive-form game: Stage 1: The health plan sets (q Ex ; eEx ; sEx ) and (q St ; eSt ; sSt ). Under cost reimbursement, the health plan commits to reimbursing Provider i’s operating cost, and sets the margin mi , i = A; B. Under prospective payment, the health plan sets the price pi , i = A; B, for each unit of service supplied by Provider i. Stage 2: Providers A and B choose qualities and cost-reduction e¤orts simultaneously. Stage 3: The health plan observes the providers’chosen qualities and e¤orts, and reports the tier that each provider belongs to. Stage 4: Consumers learn the providers’ designated tiers, and decide on obtaining services from one of 10 them. Consumers do not observe the providers’qualities and e¤orts. The health plan does observe these, and assign providers to di¤erent tiers. In Stage 4, consumers form beliefs about qualities and e¤orts based on providers’ tiers. Therefore, we characterize prefect-Bayesian equilibria of the Tired Network game. The health plan’s objective is to implement the …rst best as a perfect-Bayesian equilibrium. The next two subsections study, respectively, equilibria under cost reimbursement and prospective payment. Upon seeing 0 0 providers’ tier designations, consumers believe that qA and qB are service qualities of Providers A and B, 0 respectively. Consumers believe that utilities from Providers A and B are, respectively, vqA 0 vqB sB (1 0 x). Let x b be de…ned by vqA sA Providers A and B are, respectively, x bA = x b and x bB = 1 3.2 Cost reimbursement and …rst best 0 x b = vqB x b. sB (1 sA x and x b), then market shares of When consumers can directly observe qualities and cost e¤orts, a provider is free to choose any nonnegative pair of quality and cost e¤ort to maximize pro…t. In the conventional setup, a large literature shows that cost reimbursement leads to zero cost e¤ort and suboptimal quality. In the Tiered Network, consumers observe the providers’ tiers but not their qualities and e¤orts. Our …rst result shows that in any perfect-Bayesian equilibrium, an active provider’s choice can only be either (q Ex ; eEx ) or (q St ; eSt ). Lemma 2 Consider cost reimbursement in a Tiered Network. If a provider is assigned to the Excellent tier in a perfect-Bayesian equilibrium, its quality and cost e¤ ort are (q Ex ; eEx ). Likewise, if a provider is assigned to the Standard tier in a perfect-Bayesian equilibrium, its quality and cost e¤ ort are (q St ; eSt ). According to Lemma 2, in equilibrium, a provider must use the lowest possible quality and cost e¤ort to qualify for the Excellent or Standard tier. This is because in Stage 4 consumers only observe a provider’s designated tier, but not its quality or e¤ort choices. Consumers’beliefs on providers’qualities depend only on the tier designation, so the tier designation also determines each provider’s market share and revenue. Among combinations of quality and cost e¤ort that qualify a provider for a tier, a pro…t-maximizing provider must choose the least costly pair, one that minimizes G(q) + H(e). Hence, in equilibrium, when consumers 11 observe the providers’designated tiers, they must believe that the providers have chosen the lowest possible quality and cost e¤ort in a quali…ed tier. The health plan can utilize the tiers to incentivize optimal quality and cost-e¤ort choices. In fact, the following is an immediate consequence of Lemma 2 (and its proof is omitted). Corollary 1 Consider cost reimbursement in a Tiered network. The …rst best is implementable as a perfectBayesian equilibrium only if the health plan sets thresholds (q Ex ; eEx ) = (qA ; eA ) for the Excellent tier and (q St ; eSt ) = (qB ; eB ) for the Standard tier. We next derive the condition on the copayments for the implementation of the …rst best. Suppose that in Stage 2, Providers A and B choose equilibrium (qA ; eA ) and (qB ; eB ), respectively. In Stage 3 the health plan then reports that Provider A belongs to the Excellent tier and Provider B to the Standard tier. By Lemma 2, consumers’ demand for services from Provider A in Stage 4 is 1=2 + (1=2 )[v(qA qB ) (sEx sSt )]. To implement the …rst best, the health plan must choose sEx and sSt such that the consumer demand for provider A’s services equals x b in condition (6). Therefore, in a …rst-best equilibrium, the copayments must satisfy sEx sSt = C(qA ; eA ) C(qB ; eB ): the di¤erence between the copayments in the Excellent and Standard tiers must be equal to the di¤erence in …rst-best unit costs of the more e¢ cient Provider A and the less e¢ cient Provider B. Lemma 3 Suppose the …rst best is implementable as a perfect-Bayesian equilibrium in a Tiered Network. The lowest, nonnegative copayments are sEx = 0, sSt = C(qB ; eB ) and sEx = C(qA ; eA ) C(qB ; eB ), sSt = 0 if C(qA ; eA ) C(qA ; eA ) if C(qA ; eA ) < C(qB ; eB ), C(qB ; eB ). Our assumptions do not pin down the relative magnitude of C(qA ; eA ) and C(qB ; eB ). Lemma 1 prescribes higher quality and higher cost e¤ort for Provider A, due to its lower …xed costs, so the operating cost C(qA ; eA ) may be lower or higher than Provider B’s operating cost C(qB ; eB ). Consider a set of parameters for which C(qA ; eA ) < C(qB ; eB ). De…ne the iso-unit cost line by the combinations of q and e such that C(q; e) = C(qB ; eB ). If C(qA ; eA ) < C(qB ; eB ), then we have (qA ; eA ) located strictly above the iso-unit cost line. The dotted, upward-sloping curve in Figure 2 is the iso-unit cost curve of (the less e¢ cient) Provider B 12 Figure 2: Equilibrium tiers and copayments at the …rst-best unit cost level.8 In this case, the copayment for a Standard-tier provider is higher than an Excellent-tier provider, and the lowest nonnegative copayments are sEx = 0, sSt = C(qB ; eB ) Now if C(qA ; eA ) C(qB ; eB ), then the …rst best requires sEx = C(qA ; eA ) C(qA ; eA ). C(qB ; eB ), and sSt = 0; the copayment at the Excellent tier is higher than at the Standard tier. We now study the providers’ quality and e¤ort choices. In Stage 2, the two providers simultaneously choose qualities and e¤orts to maximize pro…ts. Because consumers’beliefs about qualities and e¤orts are constrained by Lemma 2, a pro…t-maximizing provider must choose (q Ex ; eEx ) to qualify for the Excellent tier and (q St ; eSt ) to qualify for the Standard tier. Let the tier thresholds and copayments be given by Corollary 1 and Lemma 3. Table 1 de…nes the providers’market shares in continuation games with di¤erent tier designations. For example, if Provider A chooses (qA ; eA ) and Provider B chooses (qB ; eB ), then Provider A quali…es for the Excellent tier, while Provider B quali…es for the Standard tier. Therefore, their market shares are x b and 1 x b , respectively. This shows up as the …rst row and the second column in Table 1. If they both qualify for the Excellent tier or the Standard tier, they split the market equally; these are in the …rst-row-…rst-column, and second-row-second-column cells in Table 1. 8 The iso-unit cost curve is upward sloping because C is increasing in q and decreasing in e. Furthermore, the lower-contour set of a convex function is convex, and hence the iso-unit cost curve has the shaped as drawn. 13 Provider B Provider A (qA ; eA ) 1=2; 1=2 1 x b ;x b 0; 1 (qA ; eA ) (qB ; eB ) (0; 0) (qB ; eB ) x b ;1 x b 1=2; 1=2 0; 1 (0; 0) 1; 0 1; 0 0; 0 Table 1: Equilibrium qualities, e¤orts, and market shares In a …rst-best equilibrium under cost reimbursement, Provider A must choose (qA ; eA ) and its pro…t is mA x b [G(qA )+H(eA )]. Similarly, Provider B must choose (qB ; eB ) and its pro…t is mB (1 x b ) [G(qB )+ H(eB )]. To implement the …rst best, the health plan sets margins mA and mB such that neither provider …nds a unilateral deviation pro…table. The …rst best is implementable if there are margins mA and mB that satisfy the following incentive constraints: mB (1 mA x b x b ) [G(qA ) + H(eA )] mA =2 [G(qB ) + H(eB )] mB =2 [G(qB ) + H(eB )] [G(qA ) + H(eA )]; (7) (8) and nonnegative pro…t constraints: mB (1 mA x b x b ) [G(qA ) + H(eA )] 0 (9) [G(qB ) + H(eB )] 0: (10) In a …rst-best equilibrium, Provider A chooses (qA ; eA ). The incentive constraint (7) says that Provider A should not pro…t by deviating to (qB ; eB ) and capturing 1=2 of the market (see Table 1). The incentive constraint (8) has the corresponding meaning for Provider B. The incentive constraints guarantee that Provider A picking (qA ; eA ) and Provider B picking (qB ; eB ) are mutual best responses (given equilibrium consumer beliefs). The nonnegative pro…t constraints (9) and (10) guarantee that in equilibrium both providers prefer to be active. Can we …nd margins mA and mB to implement the …rst best as a perfectBayesian equilibrium? In other words, which of the incentive and nonnegative pro…t constraints are more binding? By Lemma 1, x b > 1=2 and G(qA ) + H(eA ) > G(qB ) + H(eB ). Therefore, a su¢ ciently big mA can satisfy both (7) and (9), so Provider A has no incentive to deviate from the Excellent tier if mA is su¢ ciently 14 big. The incentive and nonnegative pro…t constraints for Provider B work di¤erently. Conditions (8) and (10) constrain mB in opposite directions. The incentive constraint (8) requires mB to be small, so that deviating to (qA ; eA ) and getting the additional (b x 1=2) consumers is unpro…table. The nonnegative pro…t constraint (10) requires mB to be big so that Provider B remains active. Combining both constraints, we obtain [G(qA ) + H(eA ) G(qB ) x b 1=2 H(eB )] mB which yields the condition in the next Proposition. [G(qB ) + H(eB )] 1 x b (11) Proposition 1 Consider cost reimbursement in a Tiered Network. Let the tier and copayment policies be given by those in Corollary 1 and Lemma 3. The …rst best is implemented as a perfect-Bayesian equilibrium if and only if G(qA ) + H(eA ) G(qB ) + H(eB ) : 2(1 x b ) (12) Inequality (12) in Proposition 1 says that in the …rst best, the total …xed cost of quality and cost e¤ort for the more e¢ cient Provider A must still be quite larger than the less e¢ cient Provider B. This is a direct consequence of the requirement that mB stays within the bounds in (11) for the …rst best. We next turn to the cost of …rst-best implementation. The health plan’s total payment to the providers in b ). Therefore, the health plan minimizes x +[mB +C(qB ; eB )](1 x a …rst-best equilibrium is [mA +C(qA ; eA )]b payment to the providers when the margins are chosen to be the smallest required for implementation. The value of mB must satisfy (11), so the minimum feasible value must be [G(qB )+H(eB )] . 1 x b The value of mA must satisfy (7) and (9). Our next result shows that the incentive constraint (7) is the more binding one. Corollary 2 Consider the implementation of the …rst best in a Tiered Network. The cost-minimizing margins are mA = mB = [G(qA ) + H(eA ) G(qB ) x b 1=2 [G(qB ) + H(eB )] : 1 x b H(eB )] (13) (14) The minimum margin mA in (13) for implementing the …rst best is obtained by a binding Provider A incentive constraint (7). Provider A’s nonnegative pro…t constraint (9) is slack. The more e¢ cient Provider 15 A must make a pro…t in a …rst-best equilibrium. This is a consequence of condition (12) in Proposition 1; this requires x b to be not much larger than 1=2, and G(qA ) + H(eA ) larger than G(qB ) + H(eB ). Condition (12) ensures that the more e¢ cient Provider A makes a positive pro…t if it deviates to (qB ; eB ). Conversely, the value of mB in (14) comes from a binding Provider B nonnegative pro…t constraint (10). Condition (12) in Proposition 1 says that at that value the incentive constraint (8) is slack: the less e¢ cient Provider B makes a negative pro…t if it deviates to (qA ; eA ). Therefore, Provider B’s pro…t is zero in the …rst-best equilibrium. 3.3 Prospective payment and …rst best We now study the implementation of the …rst best under prospective payment. Here, Providers A and B receive pA and pB , respectively, for each unit of service. Each provider fully internalizes the service unit costs and …xed costs. Because of this internalization, the speci…cation of tiers according to (q Ex ; eEx ) or (q St ; eSt ) in Lemma 2 can be relaxed. We can alternatively de…ne a tier based on a provider’s quality only. For example, a provider belongs to the Excellent tier if its quality is at least q Ex ; a provider belongs to the Standard tier if its quality is at least q St . However, for ease of exposition, we will continue to use the same tier construction as in the last subsection, as in Corollary 1. Given the same tier construction, an active provider’s choice of quality and e¤ort continue to be either (qA ; eA ) or (qB ; eB ) in a perfect-Bayesian equilibrium. We specify consumers’beliefs in the same way. That is, in a perfect-Bayesian equilibrium, consumers believe that a provider’s quality and cost e¤ort are the minimum that would qualify for its tier designation. We also let the health plan use the same copayments for …rst-best implementation as in Lemma 3 in the last subsection. The …rst best is implementable if there are prospective prices pA and pB that satisfy the incentive constraints: [pA [pB C(qA ; eA )]b x [G(qA ) + H(eA )] [pA C(qB ; eB )]=2 x b ) [G(qB ) + H(eB )] [pB C(qA ; eA )]=2 C(qB ; eB )](1 16 [G(qB ) + H(eB )] [G(qA ) + H(eA )] (15) (16) and nonnegative pro…t constraints: [pA [pB C(qA ; eA )]b x [G(qA ) + H(eA )] 0 (17) x b ) [G(qB ) + H(eB )] 0: (18) C(qB ; eB )](1 Incentive constraints (15) and (16) say that Providers A and B, respectively, prefer to achieve the Excellent and Standard tiers. The nonnegative pro…t constraints (17) and (18) ensure that both providers are active. Because x b > 1=2, both (15) and (17) can be satis…ed when pA is su¢ ciently high. A high prospective price pA encourages the more e¢ cient Provider A to attain the Excellent Tier to get a larger market share. By contrast, again because 1 x b < 1=2, Provider B’s incentive constraint (16) requires that pB be small, while the nonnegative pro…t constraint (18) requires pB to be large. By combining (16) and (18), we get the condition in the following Proposition. Proposition 2 Consider prospective payment in a Tiered Network. Let the tier and copayment policies be given by those in Corollary 1 and Lemma 3. The …rst best is implemented as a perfect-Bayesian equilibrium if and only if G(qA ) + H(eA ) G(qB ) + H(eB ) C(qB ; eB ) C(qA ; eA ) + : 2(1 x b ) 2 (19) Under prospective payment, each provider internalizes its variable cost C(q; e). This accounts for the extra term in (19) in Proposition 2 compared to (12) in Proposition 1. Provider A is more e¢ cient, and in the …rst best puts in more quality and cost e¤ort, so the comparison between variable costs of the two providers is ambiguous in general. Indeed, (19) implies (12) if and only if C(qA ; eA ) < C(qB ; eB ). When Provider A’s …rst-best variable cost is lower than Provider B’s, C(qA ; eA ) < C(qB ; eB ), the …rst best is implementable by prospective payment only if it is implementable by cost reimbursement. Symmetrically, (12) implies (19) if and only if C(qA ; eA ) C(qB ; eB ), so the …rst best is implementable by prospective payment only if it is implementable by cost reimbursement when Provider A’s variable cost is higher. The health plan pays pA x b + pB (1 x b ) to implement the …rst best. The minimum feasible value of pB must satisfy the binding nonnegative pro…t constraint (18). The minimum feasible value of pA must satisfy both the incentive constraint (15) and the nonnegative pro…t constraint (17). However, either one of these 17 may be the binding constraint. Indeed, if Provider A earns a strictly positive pro…t by deviating to the Standard tier (so that the right-hand side of (15) is strictly positive), then the nonnegative constraint (17) is slack. The converse is also true. Whether Provider A’s incentive constraint or nonnegative pro…t constraint binds depends on the comparison of the right-hand side of (15) and the left-hand side of (17). Clearly, the incentive constraint (15) implies the nonnegative pro…t constraint (17) if and only if C(qA ; eA ) + G(qA ) + H(eA ) x b C(qB ; eB ) + G(qB ) + H(eB ) : 1=2 (20) Given consumers’equilibrium responses to tier designation, the left-hand side of (20) is Provider A’s average service cost when quality and cost e¤ort qualify for the Excellent tier, while the right-hand side is the average cost when Provider A deviates to the Standard tier. Condition (20) requires x b C(qA ; eA ) to be small, and G(qA ) + H(eA ) 1=2 and C(qB ; eB ) H(eB ) to be large. G(qB ) For Provider B, condition (19) in Proposition 2 says that its incentive constraint (16) is slack. The binding nonnegative pro…t constraint (18) determines the minimum feasible value of pB . The next Corollary shows how inequality (20) determines the minimum costs of …rst-best implementation. Corollary 3 Consider the implementation of the …rst best in a Tiered Network. Suppose that the incentive constraint (15) implies the nonnegative pro…t constraint (17) (equivalent to (20) satis…ed), the costminimizing prospective price for Provider A is pA = C(qA ; eA )b x C(qB ; eB )=2 + G(qA ) + H(eA ) x b 1=2 G(qB ) H(eB ) : (21) Suppose that the nonnegative pro…t constraint (17) implies the incentive constraint (15) (equivalent to (20) violated), the cost-minimizing prospective price for Provider A is pA = C(qA ; eA ) + [G(qA ) + H(eA )] : x b (22) The cost-minimizing prospective price for Provider B is pB = C(qB ; eB ) + 18 [G(qB ) + H(eB )] : 1 x b (23) 3.4 Payment mechanism and implementation cost We now compare the implementation costs under cost reimbursement and prospective payment. Clearly, the comparison is meaningful only when the …rst best can be implemented by both mechanisms. According to Propositions 1 and 2, this is true when conditions (12) and (19) are satis…ed. The resource costs under the …rst best are predetermined, so the implementation cost comparison has to do with providers’ pro…ts. In other words, we should compare the margin-unit-cost combinations under cost reimbursement and the prospective prices. By Corollaries 2 and 3, the less e¢ cient Provider B always makes a zero pro…t, so mB + C(qB ; eB ) = pB (see also (14) and (23) in the two Corollaries). The comparison therefore boils down to the values of mA + C(qA ; eA ) and pA in the two Corollaries. By Corollary 2, the minimum margin for Provider A is in (13). By Corollary 3, the minimum prospective price for Provider A is either (21) or (22), depending on whether the incentive constraint binds or the nonnegative pro…t constraint binds. First, suppose that Provider A’s price is given by (21). Combining this and (13), we obtain (details in the proof) pA The equality says that pA [mA + C(qA ; eA )] = C(qA ; eA ) 2b x C(qB ; eB ) : 1 [mA + C(qA ; eA )] if and only if C(qA ; eA ) (24) C(qB ; eB ). Under prospective payment, when deviating from (qA ; eA ) to (qB ; eB ), Provider A internalizes the change in unit cost. This internalization is absent under cost reimbursement. Therefore, when C(qA ; eA ) C(qB ; eB ), a deviation from (qA ; eA ) to (qB ; eB ) is more pro…table under prospective payment. The health plan has to pay a higher price to keep Provider A from deviating to (qB ; eB ). Symmetrically, if C(qA ; eA ) < C(qB ; eB ), the same deviation to (qB ; eB ) is less pro…table under prospective payment. The health plan only has to pay a lower price. Second, suppose that Provider A’s price is given by (22). Here, Provider A makes a zero pro…t under prospective payment. However, according to Corollary 2, Provider A always makes a strictly positive pro…t under cost reimbursement. (The proof of that Corollary establishes that the minimum margin is obtained from a binding incentive constraint while the nonnegative pro…t constraint remains slack.) Therefore, we 19 must have pA < [mA + C(qA ; eA )], so that implementation cost is lower under prospective payment. Now, Provider A’s price being given by (22) requires the violation of (20). As the proof of the next Proposition shows, this, together with the necessary condition of implementation by prospective payment implies C(qB ; eB ) > C(qA ; eA ). Proposition 3 Suppose the …rst best can be implemented by cost reimbursement and prospective payment in a Tiered Network. The health plan minimizes the payment to providers by using cost reimbursement if and only if C(qA ; eA ) C(qB ; eB ). In this section we have compared cost reimbursement and prospective payment in Tiered Networks. We have focused on the implementation of the …rst best. In fact, our results can be applied to the implementation of other qualities, cost e¤orts, and assignments of consumers across the providers. In the …rst best, Provider A chooses higher quality and cost e¤ort, and serves more consumers than Provider B. These are the only properties used for all the lemmas, propositions, and corollaries here. If we replace qA , eA , qB , eB , and x b by qeA , eeA , qeB , eeB , and x e in these lemmas, propositions, and corollaries, the statements remain correct when qeA > qeB , eeA > eeB , and x e > 1=2. This can be veri…ed by inspection. Cost and quality e¢ ciencies di¤er among the providers, and this yields an asymmetric …rst best. Suppose that, for whatever reason, a health plan wants to implement an allocation di¤erent from the …rst best. Our results continue to hold as long as the allocation exhibits the same kind of asymmetry, namely, the more e¢ cient provider puts in more quality and cost e¤ort, and serves more consumers. It is hard for us to imagine situations in which any health plan would want to implement allocations that favor the less e¢ cient provider. Therefore, we do not consider implementation of these uninteresting allocations. 4 4.1 Value-based Network Network structure and extensive form In this section we consider the implementation of the …rst best in Value-based Networks. Here, the health plan constructs a value index for each provider based on its quality and cost-reduction e¤ort, and continues 20 to pay the providers by either cost reimbursement or prospective payment. A value index is a weighted average of a provider’s quality and cost e¤ort. Suppose that Provider i, i = A; B, chooses quality qi and cost e¤ort ei . As in the previous section, these choices are noncontractible, but the health plan can observe them, and disclose some information about them. The value index for Provider i, i = A; B, is Ii ( i ) (1 i )ei , B, where 0 i 1, is the weight on quality. The health plan can commit to the weights i qi + A and as well as payment parameters and consumer copayments before the providers make decisions. The extensive form for the Value-based Network game is as follows. Stage 1: The health plan sets ( A; B ) and copayments (sA ; sB ). Under cost reimbursement, the health plan sets the margin mi , i = A; B, and commits to reimbursing Provider i’s operating cost. Under prospective payment, the health plan sets the price pi , i = A; B, for each unit of service supplied by Provider i. Stage 2: Providers A and B choose qualities and cost-reduction e¤orts simultaneously. Stage 3: The health plan observes the providers’chosen qualities and e¤orts, and reports the value index Ii ( i ) = i qi + (1 i )ei of each Provider i = A; B. Stage 4: Consumers learn the providers’value indexes (but not providers’qualities and cost e¤orts), and decide between obtaining services from Provider A and Provider B. We study perfect-Bayesian equilibria under cost reimbursement and prospective payment in the next two subsections. 4.2 Cost reimbursement and …rst best In Tiered Networks, a health plan would only announce if a provider quali…es for one of two possible tiers. In Value-based Networks, a provider’s value index can take on any nonnegative number. Each level of index Ii ( i ) can be achieved by many combinations of qi and ei . Given such indexes, what do consumers believe about providers’qualities and cost e¤orts in a perfect-Bayesian equilibrium? Suppose that equilibrium value indexes of Providers A and B are, respectively, IA ( 21 A) and IB ( B ), and that in equilibrium, Provider A serves x b of consumers while Provider B serves the rest. The next Lemma shows that in any perfect-Bayesian equilibrium, Provider i must choose the pro…t-maximizing qi and ei to achieve index Ii ( i ). Lemma 4 Consider cost reimbursement in a Valued-based Network. Suppose that in equilibrium Provider A serves x b of consumers while Provider B serves the rest. Equilibrium qualities and cost e¤ orts (qA ; eA ) and (qB ; eB ) must solve max mA x b 0 ;e0 qA A subject to 0 A qA 0 [G(qA ) + H(e0A )] 0 A )eA + (1 = A qA + (1 A )eA and max mB (1 0 ;e0 qB B subject to 0 B qB + (1 x b) 0 [G(qB ) + H(e0B )] 0 B )eB = B qB + (1 B )eB : Hence, (qi ; ei ), i = A; B, satisfy G0 (qi ) = H 0 (ei ) 1 i : (25) i As in Tiered Networks, a provider will attempt to achieve a level of the value index in the most pro…table way. In equilibrium, consumers must believe that pro…t-maximizing qualities and e¤orts are used to achieve an index. Once providers’index levels are given, demands, and hence revenues, are …xed. Then Lemma 4 implies that in equilibrium qi and ei must minimize the total …xed cost to achieve index Ii ( i ). Condition (25) is the solution to this minimization problem. It says that the ratio of marginal …xed cost G0 (qi )=H 0 (ei ) must be set equal to the ratio of quality and cost-e¤ort weights i =(1 i ). In other words, the marginal contributions of quality and cost e¤ort to achieve an index must be in-line to their respective marginal costs. Figure 3 illustrates Provider i’s cost-minimization problem. The downward sloping straight line contains (q; e) pairs that have the same level of value index Ii with the weight on quality being set at i. The iso-cost curve plots (q; e) pairs that have the same total …xed cost; it is concave to the origin because G and H are strictly convex. The tangency point (qi ( i ); ei ( i )) shows the cost-minimizing quality and cost e¤ort at Ii ( i ). For a …xed i, the dotted “expansion path” from the origin describes the cost-minimizing qualities 22 Figure 3: Cost-minimizing quality and cost e¤ort and cost e¤orts that achieve di¤erent levels of value index Ii ( i ) in a cost-minimizing way. When the value of i changes, the entire expansion path will pivot around the origin. For example, if i increases so that more weight is given to quality, then more quality will be chosen to achieve any given level of the index, so the dotted expansion path will rotate in a clockwise direction. To implement the …rst best, the health plan has to choose index weights A and B so that the …rst best is on the providers’expansion paths, or equivalently (25) holds at the …rst-best qualities and e¤orts. Corollary 4 Consider cost reimbursement in a Value-based Network. The …rst best is implementable as a perfect-Bayesian equilibrium only if the health plan sets index weights A = 0 G0 (q G (qB ) . 0 B ) + H (eB ) G0 (qA ) and G0 (qA ) + H 0 (eA ) B = The health plan also sets copayments sA and sB in Stage 1. Suppose that in Stage 2, Providers A and B choose (qA ; eA ) and (qB ; eB ), respectively. By Lemma 4, consumers can correctly infer qualities from the value indexes. Their demand for services from Provider A is 1=2 + (1=2 )[v(qA To implement the …rst-best consumer allocation, the copayment di¤erential sA qB ) (sA sB )]. sB has to internalize the di¤erence in unit costs C(qA ; eA ) C(qB ; eB ). Therefore, the lowest, nonnegative sA and sB are, respectively, identical to sEx and sSt in the optimal Tiered Network (see Lemma 3). Consumers’copayments remain the same whether the …rst best is implemented under Tiered or Value-based Networks; it follows that they obtain 23 exactly the same utilities in each network. Next, we derive the margins mA and mB for …rst-best implementation. By Lemma 4, each provider has to choose among combinations of qualities and cost e¤orts under constraint (25), which describe what can arise in equilibrium. Given mA and mB , the equilibrium in Stage 2 is given by the solutions of the two following programs: max mA qA ;eA 1 v(qA + 2 qB ) (sA sB ) 2 G0 (qA ) subject to = H 0 (eA ) 1 [G(qA ) + H(eA )] (26) A A and max mB qB ;eB 1 v(qB + 2 qA ) (sB sA ) 2 G0 (qB ) subject to = H 0 (eB ) 1 [G(qB ) + H(eB )] B (27) : B In Stage 1, the health plan must choose mA and mB so that the …rst best is a continuation equilibrium. Proposition 4 Consider cost reimbursement in a Valued-based Network. Let the index weights be given by Corollary 4 and let the copayments satisfy sA sB = C(qA ; eA ) C(qB ; eB ). The …rst best is implemented as a perfect-Bayesian equilibrium if and only if mA = mB = 1 v=2 v=2 G0 (qA ) + H 0 (eA ) G0 (qB ) + H 0 (eB ) d ln G0 (qA ) d ln H 0 (eA ) d ln G0 (qB ) : d ln H 0 (eB ) (28) (29) Proposition 4 presents a unique pair of margins (mA ; mB ) for …rst-best implementation. To obtain these margins, we begin with the constraints in (26) and (27). Each implicitly de…nes the equilibrium ei , i = A; B, as a function of qi , say ei (qi ) (and this is illustrated by the dotted expansion path in Figure 3). The slope of this function, 1 i i G00 (qi ) (derived in the Appendix), describes how, in equilibrium, ei changes with H 00 (ei ) respect to qi . Consider Provider A. To induce the …rst-best quality and e¤ort, the health plan has to set the margin mA such that the provider’s marginal bene…t of quality investment, mA v=2 , is equal to the marginal …xed cost due to investment, G0 (qA ) + H 0 (eA ) e0A (qA ), evaluated at the …rst best. The expressions for mA in Proposition 4 is obtained by solving this equation. The expression for mB is derived in the same way. 24