Tiered and Value-based Health Care Networks Appendix

Appendix Proof of Lemma 1: We obtain equations (2) to (6) by setting the …rst-order derivatives of (1) with respect to qA , qB , eA , eB , and x b, to 0. These are necessary conditions for the maximization of (1). We now show that qA > qB , eA > eB , and x b > 1=2. By symmetry, if was set at 1, then we would have qA = qB , eA = eB , x b = 1=2. Therefore, it is su¢ cient to show that qA , eB , and x b are qB , eA , monotone increasing in . Our assumptions on C and G and H are laid out in Subsection 2.1. However, for the purpose here, we would strengthen them by restricting the sets of feasible quality and cost e¤ort, q and e for each provider to those values where v Cq (q; e) > 0. According to the …rst-order conditions (2) and (3), there cannot be any solution at which v Cq (q; e) < 0. Such a restriction, v Cq (q; e) > 0, does not a¤ect the …rst best. Now we can apply Theorems 5 and 6 in Milgrom and Shannon (1994), which say that solutions, qA , qB , eA , eB , and x b , that maximize (1) are monotone increasing in derivatives of qA , qB , eA , @2W =0 @ @qA eB , x b, and are nonnegative. Indeed, we have: @2W = G0 (qB ) > 0 @ @qB @2W = H 0 (eB ) > 0 @ @eB @2W = @qA @eA when all the pairwise cross-partial x bCqe (qA ; eA ) @2W =0 @ @eA @2W =0 @qA @qB @2W =0 @ @b x 0 @2W =0 @qA @eB @2W =0 @qB @eA @2W = @qB @eB @2W =0 @eA @eB @2W = @eA @b x (1 @2W =v @qA @b x x b)Cqe (qB ; eB ) Ce(qA ; eA ) > 0 0 @2W =v @qB @b x @2W = @eB @b x Cq (qA ; eA ) > 0 Cq (qB ; eB ) > 0 Ce(qB ; eB ) > 0: Proof of Lemma 2: Suppose that the Lemma is false, so suppose that in a perfect-Bayesian equilibrium, Provider i is assigned to the Excellent tier and (q Ex ; eEx ) 6= (qi ; ei ). By the de…nition of the Excellent tier, we must have qi > q Ex , ei > eEx , or both. Let the equilibrium margin and market share of provider i be mi and x bi , respectively. Under cost reimbursement, Provider A’s equilibrium pro…t is mi x bi G(qi ) H(ei ). Now let Provider i deviate from (qi ; ei ) to (q Ex ; eEx ). Provider i still belongs to the Excellent tier, which means that consumers’beliefs about Provider A’s quality remain the same, and so Provider A’s market share must remain at x bi . But because G and H are strictly increasing, we have mi x bi G(q Ex ) H(eEx ) > mi x bi G(qi ) H(ei ), which says that the deviation is pro…table, a contradiction. The proof for the Standard tier is similar, and omitted. Proof of Lemma 3: The Lemma follows immediately from the equation sEx C(qB ; eB ) and the inequalities sEx 0 and sSt sSt = C(qA ; eA ) 0. Proof of Proposition 1: Rearranging inequalities (7) to (10), we have mA mB mA mB G(qA ) + H(eA ) G(qB ) H(eB ) x b 1=2 [G(qA ) + H(eA ) G(qB ) H(eB )] x b 1=2 [G(qA ) + H(eA )] x b [G(qB ) + H(eB )] : 1 x b (34) (35) (36) (37) A su¢ ciently big value of mA can always be chosen to satisfy both (34) and (36), so the …rst best is implementable if and only if there is an mB satisfying both (35) and (37). This is equivalent to [G(qA ) + H(eA ) G(qB ) x b 1=2 which simpli…es to (12) in the Proposition. H(eB )] [G(qB ) + H(eB )] ; 1 x b (38) Proof of Corollary 2: The minimum value for mB has already been derived just before the presentation of the Corollary. Consider Provider A. The value of mA must satisfy both (34) and (36) (see the proof of Proposition 1). The minimum value for mA must be the larger of the right-hand side expressions of (34) and (36). We now show that the right-hand side expression in (34) is the larger one. The di¤erence between them, after simpli…cation, is: = G(qA ) + H(eA ) G(qB ) H(eB ) [G(qA ) + H(eA )] x b 1=2 x b [G(qA ) + H(eA )]=2 [G(qB ) + H(eB )]b x : x b (b x 1=2) Since the denominator is always positive (b x > 1=2), the sign of the above is that of the numerator. That is, the right-hand side of (34) is larger than the right-hand side of (36) if and only if G(qA ) + H(eA ) > 2b x . G(qB ) + H(eB ) Now by (12), it is su¢ cient to show that 1 2(1 = x b ) > 2b x . For this, we calculate 1 2b x 2(1 x b ) (1 2b x )2 > 0: 2(1 x b ) Hence, the lowest possible …rst-best equilibrium mA is the one in (13). Proof of Proposition 2: Rearranging inequalities (15) to (18), we get pA pB pA pB C(qA ; eA )b x C(qB ; eB )=2 + G(qA ) + H(eA ) G(qB ) H(eB ) x b 1=2 C(qA ; eA )=2 C(qB ; eB )(1 x b ) + [G(qA ) + H(eA ) G(qB ) H(eB )] x b 1=2 [G(qA ) + H(eA )] C(qA ; eA ) + x b [G(qB ) + H(eB )] C(qB ; eB ) + : 1 x b (39) (40) (41) (42) Conditions (39) and (41) can be satis…ed by setting pA to a su¢ ciently large value. Therefore, the …rst best is implementable only if there exists pB that satis…es (40) and (42). This is equivalent to C(qA ; eA )=2 x b ) + [G(qA ) + H(eA ) x b 1=2 C(qB ; eB )(1 which can be rewritten as [G(qA ) + H(eA ) G(qB ) x b 1=2 H(eB )] + C(qA ; eA )=2 x b G(qB ) H(eB )] C(qB ; eB )(1 1=2 x b ) C(qB ; eB ) + [G(qB ) + H(eB )] 1 x b [G(qB ) + H(eB )] + C(qB ; eB ): 1 x b Next, we note that inequality (12) in Proposition 1 follows from inequality (38) in the proof of that Proposition. By inspection, we only have to simplify the second term on each side of the above. Indeed, after the simpli…cation, we obtain (19). Proof of Corollary 3: We obtain (23) by setting expression (18) as an equality. For Provider A, the minimum value of pA is the larger of the right-hand side expressions of (15) and (17). The di¤erence between the two expressions is C(qA ; eA )b x = C(qB ; eB )=2 + G(qA ) + H(eA ) G(qB ) H(eB ) C(qA ; eA ) x b 1=2 1 G(qA ) + H(eA ) G(qB ) + H(eB ) C(qA ; eA ) + C(qB ; eB ) : 2b x 1 x b 1=2 [G(qA ) + H(eA )] x b This expression is positive if and only if condition (20) is satis…ed. When condition (20) holds, the minimum pA must satisfy (15) as an equality. When condition (20) fails, the minimum pA must satisfy (17) as an equality. We thereby obtain expressions (21) and (22) for the minimum pA . Proof of Proposition 3: We have shown that mB + C(qB ; eB ) = pB before the presentation of the Proposition. We now compare the minimum values of mA + C(qA ; eA ) and pA . There are two cases. Case 1: The prospective price is given by (21) in Corollary 3 (which is equivalent to inequality (20) satis…ed). Using expressions (13) and (21), we have pA = = [mA + C(qA ; eA )] C(qA ; eA )b x C(qB ; eB )=2 + G(qA ) + H(eA ) G(qB ) H(eB ) x b 1=2 G(qA ) + H(eA ) G(qB ) H(eB ) C(qA ; eA ) x b 1=2 C(qA ; eA ) C(qB ; eB ) ; 2b x 1 which is positive if and only if C(qA ; eA ) C(qB ; eB ). Case 2: The prospective price is given by (22), which is equivalent to inequality (20) failing to hold: C(qA ; eA ) + which simpli…es to G(qA ) + H(eA ) G(qB ) + H(eB ) < C(qB ; eB ) + ; x b 1=2 2b x [G(qB ) + H(eB )] + x b [C(qB ; eB ) C(qA ; eA )] > G(qA ) + H(eA ): (43) In this case, Provider A makes a positive pro…t under cost reimbursement and zero pro…t under prospective payment. Hence we have pA < mA + C(qA ; eA ). Now from Proposition 2, …rst-best implementation requires (19). This condition and (43) can be satis…ed simultaneously only if C(qB ; eB ) C(qA ; eA ) > (1 (2b x where the last inequality follows from x b > 1=2 and We conclude that pA 2b x )2 [G(qB ) + H(eB )] > 0; 1= )(1 x b ) > 1. mA + C(qA ; eA ) if and only if C(qA ; eA ) C(qB ; eB ). Proof of Lemma 4: Suppose that in equilibrium Provider A’s market share is x b. Let equilibrium (qA ; eA ) achieve index AqA+ (1 A)eA. Next, suppose that Provider A deviates to any (q0 ; e0A) where A 0 Aq + A (1 0 A )eA = A qA + (1 A )eA . Consumers must continue to believe that the quality and cost e¤ort remain 0 at qA and eA . Provider A’s demand remains at x b. Moreover, mA x b [G(qA )+H(e0A )] mA x b [G(qA )+H(eA )], 0 0 with a strict inequality when (qA ; e0A ) 6= (qA ; eA ) due to the strict convexity of G and H. Hence (qA ; e0A ) is not a pro…table deviation and (qA ; eA ) has the property as stated in the Lemma. The proof for equilibrium (qB ; eB ) follows symmetrically. Finally, (25) are the …rst-order conditions of the two pro…t maximization programs in the Lemma. Equilibrium (qi ; ei ), i = A; B, must satisfy (25). Proof of Proposition 4: Suppose that the quality and cost e¤ort of Provider B is (qB ; eB ). Consider the pro…t-maximization problem of Provider A. By Corollary 4, the …rst-best quality and e¤ort satisfy G0 (qA ) = H 0 (eA ) 1 A and hence can be Provider A’s optimal choice. By Lemma 4, we use (25) to de…ne eA as A a function of qA , say eA (qA ). Then we calculate its derivative: e0A (qA ) = 1 A A G00 (qA ) > 0: H 00 (eA ) (44) We then rewrite Provider A’s constrained pro…t maximization program as max mA qA 1 v(qA + 2 qB ) (sA 2 sB ) [G(qA ) + H(eA (qA ))]: The …rst-order condition is mA v = G0 (qA ) + H 0 (eA )e0A (qA ): 2 (45) The margin mA for …rst-best implementation must satisfy (45) at qA = qA and qB = qB . This margin is unique because both G0 (qA ) and H 0 (eA )e0A (qA ) are increasing in qA . We obtain the expression for mA in the Proposition by using equations (25), (44), and the identity d ln f = f 0 =f to simplify (45). The derivation of mB follows the same steps. Given the two margins in the Proposition, no unilateral deviation from the …rst best is pro…table. Moreover, the consumer demand for services at Provider A is 1 v(qA + 2 qB ) (sA 2 sB ) = 1 v(qA + 2 qB ) [C(qA ; eA ) 2 Hence, the …rst best is implemented as a perfect-Bayesian equilibrium. C(qB ; eB )] x b : Proof of Lemma 5: Suppose in equilibrium Provider A serves x b of consumers. Let the equilibrium prospective price and index weight be pA , and A, respectively. Equilibrium quality and cost e¤ort (qA ; eA ) must solve max [pA 0 ;e0 qA A 0 C(qA ; e0A )]b x 0 [G(qA ) + H(e0A )] s.t. 0 A qA + (1 0 A )eA = A qA + (1 A )eA : 0 0 After simplifying the …rst-order conditions, and setting qA = qA , qB = qB , we have (1 A) : 0 (q ) + x [G b C A A q (qA ; eA )] x bCe (qA ; eA ) = H 0 (eA ) (46) Let x b=x b and qA = qA . Condition (46) is identical to (4), which implicitly de…nes eA , only if The proof that the …rst best is implementable only if B A = 1. = 1 follows the same steps. Proof of Proposition 5: Let the quality and cost e¤ort of Provider B be (qB ; eB ). By Lemma 5, A = 1 and hence consumers perfectly learn qA from the value index. Provider A chooses qA and eA to maximize [pA C(qA ; eA )] 1 v(qA + 2 qB ) (sA 2 sB ) [G(qA ) + H(eA )]: The …rst-order conditions with respect to qA and eA are [pA where x b 1 v(qA + 2 qB ) (sA 2 C(qA ; eA )] sB ) v 2 Cq (qA ; eA )b x = G0 (qA ) (47) Ce (qA ; eA )b x = H 0 (eA ); (48) . Conditions (4) and (48) are identical at qA = qA and eA = eA . To obtain the optimal prospective price pA , consider conditions (2) and (47). They are identical if and only if x b [v Cq (qA ; eA )] = [pA C(qA ; eA )] v 2 Cq (qA ; eA )b x : Simplifying further, we get pA = = = 2 x b + C(qA ; eA ) 2 1 v(qA + 2 qB ) + C(qB ; eB ) + v(qA [C(qA ; eA ) 2 C(qB ; eB )] + C(qA ; eA ) qB ): Given this value of pA , (qA ; eA ) is the unique solution to (47) and (48). The optimal pB is obtained by the same steps. Finally, the demand for Provider A’s services is 1 v(qA + 2 qB ) (sA 2 sB ) = 1 v(qA + 2 qB ) [C(qA ; eA ) 2 C(qB ; eB )] x b : This completes the …rst-best implementation. Proof of Proposition 6: First, in the proof of Proposition 4 we have shown that e0A (qA ) = d ln G0 (qA ) . d ln H 0 (eA ) Using this identity, we can rewrite the …rst-order condition (45) under cost reimbursement as mA v G0 (qA ) d ln G0 (qA ) 2 = : 0 H (eA ) d ln H 0 (eA ) (49) Second, combining …rst-order conditions (47) and (48) under prospective payment, we have [pA v 2 H 0 (eA ) C(qA ; eA )] G0 (qA ) Cq (qA ; eA ) : Ce (qA ; eA ) = (50) By inspecting (49) and (50), we conclude that pA > mA +C(qA ; eA ) if and only if The proof that pB > mB + C(qB ; eB ) if and only if Cq (qA ; eA ) d ln G0 (qA ) > . Ce (qA ; eA ) d ln H 0 (eA ) Cq (qB ; eB ) d ln G0 (qB ) > follows the same steps. Ce (qB ; eB ) d ln H 0 (eB ) Proof of Proposition 7: Suppose that condition (30) for Provider A is violated. We now show that the health plan incurs a lower payment to providers by using Tiered Networks. First, because Provider B’s nonnegative constraint is only binding in Tiered Networks, the health plan always minimizes the payment to Provider B by using Tiered Networks, irrespective of the choice of the payment system. Next, consider Provider A. By Proposition 6, the health plan minimizes the payment to Provider A by using prospective payment in Value-based Networks if the Provider’s cost functions violate condition (30). Now compare Provider A’s gross pro…t margin under prospective payment in Value-based Networks, pA C(qA ; eA ), to its cost margin under cost reimbursement in Tiered Networks, mA . By Corollary 2 and Proposition 5, the value of pA + v(qA qB ) C(qA ; eA ) is bigger if and only if [C(qA ; eA ) C(qB ; eB )] > Using the de…nition of x b in (6) and noting that x b [G(qA ) + H(eA ) x b 1=2 = v(qA qB ) G(qB ) 1=2 [C(qA ; eA ) 2 H(eB )] : C(qB ; eB )] , we can rewrite the inequality as or x b fv(qA x b [vqA qB ) [C(qA ; eA ) C(qA ; eA )] C(qB ; eB )]g > G(qA ) + H(eA ) [G(qA ) + H(eA )] > x b [vqB C(qB ; eB )] G(qB ) H(eB ) [G(qB ) + H(eB )]: (51) By Lemma 1, the values of qA and eA are chosen to maximize x b [vqA C(qA ; eA )] [G(qA ) + H(eA )] and (qA ; eA ) 6= (qB ; eB ). Because of the strict convexity of C, G, and H, inequality (51) holds. Hence the health plan incurs a lower payment to Provider A by using cost reimbursement in Tiered Networks, compared to using either cost reimbursement or prospective payment in Value-based Networks. This completes the proof that the health plan minimizes the payment to providers by using Tiered Networks when condition (30) for Provider A is violated. Finally, in the discussion after the statement of the Proposition, we show by an example that if condition (30) for Provider A is satis…ed, the health plan may minimize the payment to providers by using either Tiered or Value-based Networks. This completes the proof of the Proposition.