Tiered and Value-based Health Care Networks Prospective Payment and First Best

4.3 Prospective payment and …rst best We now study …rst-best implementation under prospective payment. The provider payment literature has shown that health plans and providers have aligned cost-e¢ ciency incentives under complete information. Our next result says that when the health plan constructs value indexes under prospective payment, it assigns zero weight to cost e¤ort in order to implement the …rst best. Our result rea¢ rms those in the extant literature. Lemma 5 Consider prospective payment in a Value-based Network. The …rst best is implementable as a perfect-Bayesian equilibrium only if the health plan sets A = B = 1. Under prospective payment, a provider fully internalizes the bene…t of cost reduction. There is no need for extra incentive for cost e¤ort. In other words, simply ignoring cost e¤ort in the value index, disclosing the quality, is su¢ cient. Under prospective payment, a value index should put all weights on quality. In the last subsection, we have established that consumers have to pay the same minimum copayments whenever the …rst best is implementable. But as the next Proposition shows, the optimal prospective prices and cost margins in Value-based Networks are chosen to address distinctive incentive problems. Proposition 5 Consider prospective payment in a Value-based Network. Let the index weights be B sB = C(qA ; eA ) = 1 and the copayments satisfy sA A = C(qB ; eB ). The …rst best is implemented as a perfect-Bayesian equilibrium if and only if pA = + C(qB ; eB ) + v(qA qB ) pB = + C(qA ; eA ) + v(qB qA ): Because the providers internalize the costs and bene…ts of e¤orts, their e¤ort incentives are aligned with the health plan’s. We only need to choose prospective prices to align providers’ quality incentives. Indeed, there is a basic divergence between a provider’s private quality bene…t and social quality bene…t. For example, if Provider A raises qA by one unit, its gross pro…t increases by [pA pA C(qA ; eA ) is the gross pro…t margin and where C(qA ; eA )] v , where 2 v is the provider’s gain in market share. However, the 2 increase in quality also bene…ts Provider A’s current consumers so the increase in social bene…t is vb x. This divergence of incentives can be likened to the problem of monopolistic quality provision in Spence (1975). To implement the …rst best, the health plan must choose prices so that a provider’s marginal pro…t due to quality is equal to the marginal social bene…t. The expressions for pA and pB in Proposition 5 are chosen to do that. Speci…cally, the optimal prospective price of a provider is increased when demand becomes less responsive to quality, or when the rival provider has a higher unit cost at the …rst best. Moreover, Provider A is rewarded for its higher average social bene…t v(qA B is reduced for its lower average social bene…t v(qB 4.4 qB ) > 0, whereas the prospective price for Provider qA ) < 0. Payment mechanism and implementation cost We now study implementation costs under the two payment systems. Unlike in Tiered Networks where only Provider A can make positive pro…ts, both providers earn positive pro…ts in Value-based Networks. Implementation costs take the form of provider pro…ts. Provider i’s, i = A; B, pro…t margin under prospective payment is pi C(qi ; ei ), and mi under cost reimbursement, so it is su¢ cient to compare these two values. In the next proposition we compare the values of mi + C(qi ; ei ) and pi . Proposition 6 For Value-based Networks, Provider i earns a higher pro…t under prospective payment than cost reimbursement, pi > mi + C(qi ; ei ), if and only if Cq (qi ; ei ) d ln G0 (qi ) : > Ce (qi ; ei ) d ln H 0 (ei ) (30) The comparison between a provider’s pro…ts under cost reimbursement and prospective payment depends on the curvatures of the unit cost C and the …xed costs G and H at the …rst best. Figure 4 illustrates an example where inequality (30) in Proposition 6 holds. The solid curve in the diagram is the iso-unit cost curve of Provider i at the …rst best (see the discussion following Lemma 3). By the implicit function theorem, the slope of this curve is Cq (qi ; ei ) at (qi ; ei ). The dashed curve in Figure 4 plots the combinations of Ce (qi ; ei ) …xed-cost minimizing (qi ; ei ) that we have introduced in Figure 3. There, we also have shown that under Proposition 4, the slope of the dashed curve is 1 i i (qi ; ei ). G00 (qi ) d ln G0 (qi ) at (qi ; ei ) and can be written as at 00 H (ei ) d ln H 0 (ei ) Figure 4: Comparison of providers’pro…ts Under prospective payment, the …rst-best pi is set to make Provider i’s iso-pro…t (net of unit cost) line tangent to the solid curve at (qi ; ei ). Similarly, under cost reimbursement, the …rst-best mi is set such that Provider i’s iso-pro…t line tangents to the dashed curve at (qi ; ei ). In Figure 4 the inequality (30) is satis…ed, so the solid curve is steeper than the dashed curve at (qi ; ei ). In other words, the …rst-best cost e¤ort is more costly to Provider i under prospective payment. In order to implement the …rst best, the health plan has to o¤er a higher price under prospective payment to make Provider i’s iso-pro…t curve steeper. This means that pi is higher than mi + C(qi ; ei ). Because the comparison between pi and mi + C(qi ; ei ) depends on the curvatures of the cost functions at (qi ; ei ), it is possible that Provider A earns a higher pro…t under cost reimbursement, but Provider B earns a higher pro…t under prospective payment, and vice versa. 5 Optimal network In the previous two sections, we have studied how the …rst best can be implemented in Tiered and Valuebased Networks. By Propositions 1 and 2, the …rst best can be implemented in Tiered Networks only if either condition (12) or (19) is satis…ed. On the other hand, by Propositions 4 and 5, the …rst best can always be implemented in Value-based Networks. This di¤erence is due to the interaction between network design and providers’nonnegative pro…t constraints. In Tiered Networks, an active provider can only choose between (qA ; eA ) and (qB ; eB ) (see Lemma 2 and Corollary 4). Conditions (12) and (19) are necessary conditions for the less e¢ cient Provider B to make a nonnegative pro…t at (qB ; eB ). In Value-based Networks, an active provider can choose many pairs of qualities and cost e¤orts. Provider payments are chosen to satisfy the providers’incentive constraints, while providers’nonnegative pro…t constraints never bind. When the …rst best can be implemented in both Tiered and Value-based Networks, Provider B’s nonnegative pro…t constraint binds in Tiered Networks only. However, the binding constraint does not imply that the total implementation cost must be lower in Tiered Networks. To compare total implementation cost, we have to consider the total pro…ts of Providers A and B under di¤erent networks and payment mechanisms. We now make this comparison assuming conditions (12) and (19) are satis…ed. Proposition 7 Suppose that the …rst best can be implemented in both Tiered and Value-based Networks. If condition (30) for Provider A is violated, the health plan pays a lower cost to providers by using Tiered Networks. On the other hand, if condition (30) for Provider A holds, the health plan may pay a lower cost to providers by using either Tiered or Value-based Networks. First, consider the case where condition (30) for Provider A is violated. By Proposition 6, in Valuebased Networks the health plan pays Provider A less by using prospective payment than by using cost reimbursement. Now compare Provider A’s gross pro…t margin pA C(qA ; eA ) in Value-based Networks to its cost margin mA in Tiered Networks. In the Appendix, we show that the latter has a smaller value if and only if x b [vqA C(qA ; eA )] [G(qA ) + H(eA )] > x b [vqB C(qB ; eB )] [G(qB ) + H(eB )]: However, this inequality always holds because in the …rst best, qA and eA maximize x b [vqA C(qA ; eA )] [G(qA ) + H(eA )]. Therefore, Provider A earns a lower pro…t under cost reimbursement in Tiered Networks, compared to either cost reimbursement or prospective payment in Value-based Networks. Because Provider B also earns a lower pro…t in Tiered Networks, Value-based Networks are dominated by cost reimbursement in Tiered Networks in terms of implementation cost. The health plan’s optimal choice is between cost reimbursement and prospective payment in Tiered Networks, and the tradeo¤ is fully characterized by Proposition 3. When condition (30) for Provider A holds, the health plan pays less to Provider A by cost reimbursement rather than by prospective payment in Value-based Networks. We now use an example to illustrate that in this case, the health plan may pay a lower cost to providers by using either Tiered or Value-based Networks. Example 1 Suppose that the …xed costs of quality and e¤ort are quadratic, that is, let G(q) = Gq 2 and H(e) = He2 , where G and H are some positive parameters. Using (13) and (14), we can express the total cost margin, mA x b + mB (1 x b ), in Tiered Networks as GqA2 + HeA2 x b GqB2 1=2 HeB2 x b + GqB2 + HeB2 (1 1 x b x b ): (31) x b ): (32) Similarly, using (28) and (29), we can express the total cost margin in Value-based Networks as (2=qA )(GqA2 + HeA2 ) x b + v=2 (2=qB )(GqB2 + HeB2 ) (1 v=2 Upon simpli…cation, the value of (31) is smaller than the value of (32) if 1 x b 1=2 (GqA2 + HeA2 ) < 2 and 1 (2=qA )(GqA2 + HeA2 ); v=2 (33) but the reverse can be true if inequality (33) fails to hold.9 In general, the comparison between the cost margins in Tiered and Value-based Networks depends on the values of G(q), H(e), and their …rst-order and second-order derivatives at the …rst best. The quadratic …xed costs in Example 1 allow us to simplify the comparison, and focus on the incentive properties of network design. On the left-hand side of (33), x b 1=2 is the demand response when Provider A raises quality from qB to qA in Tiered Networks, and GqA2 + HeA2 is the total …xed cost of Provider A. On the right-hand side of (33), v is the demand response when Provider A raises qA by one unit in Value-based Networks, 2 whereas (2=qA )(GqA2 + HeA2 ) is the equivalent of the …rst-best marginal …xed costs under the quadratic costs assumption. Example 1 shows that when condition (30) for Provider A holds, neither Tiered nor 9 Subtract (31) from (32) and rearrange terms, the value of (31) is smaller than the value of (32) if and only if 2=qA 2=qB (b x 1=2) x b 1 1 [GqA2 +HeA2 ]b x + [GqB2 +HeB2 ](1 x b ) > 0. Because v=2 x b 1=2 v=2 x b 1=2 (1 x b ) (b x 1=2) x b qA < qB and 1 for any x b < 1 if 2, the value of the …rst square-bracketed term is smaller (1 x b ) than the value of the second square-bracketed term. Moreover, the value of the …rst square-bracketed term is positive if inequality (33) is satis…ed. Valued-based Network must dominate the other in terms of implementation cost. In the example, a network design is more likely to have a lower implementation cost if it can generate a stronger demand response. 6 Conclusion In this paper, we have studied health plan designs. Our contribution is threefold. First, we introduce Tiered and Value-based Networks. A health plan uses one of these two disclosure methods to inform consumers about providers’qualities and costs, as well as to incentivize providers to choose qualities and cost e¤orts. Second, we pair each disclosure method with a payment mechanism, either cost reimbursement or prospective payment, for the implementation of e¢ cient qualities and cost e¤orts. Third, we set consumer copayments to implement the e¢ cient allocations of consumers across providers. We have derived conditions for each type of networks to implement the …rst best. Where the …rst best can be implemented, we compare the associated costs in di¤erent networks. Our approach illustrates the multifaceted nature of health plan designs. The previous literature has contrasted provider incentives due to cost reimbursement and prospective payment. We view payment mechanism only as a component in the overall health plan design. Together, payment mechanisms for providers and information mechanism for consumers determine how providers choose qualities and cost e¤ort, and how consumers choose providers. We show how health plans can exploit this interaction. In fact, we show how health plans may implement …rst best in various ways, which may require di¤erent implementation costs. We have looked at payment policies and information-based health plans with a broad perspective. While …nancial incentives are certainly important, our innovation is to supplement provider …nancial incentives with information disclosure to consumers. The simplistic way of disclosing all relevant information to consumers misses the point. Optimal network design considers optimal information disclosure. Both Tiered and Valuebased Networks add to the power of …nancial incentives. Information-based policies may prove powerful in many aspects of health care deliveries.