Online Supplements to Performance-Based Contracts for Outpatient Medical Services

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1 Jing, Png nd Svin: Performnce-bsed Contrcts Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 1 Online Supplements to Performnce-Bsed Contrcts for Outptient Medicl Services Appendix A: Nottion Houyun Jing Judge Business School, University of Cmbridge Cmbridge, CB2 1AG, United Kingdom, h.jing@jbs.cm.c.uk Zhn Png Lncster University Mngement School Lncster, LA1 4YW, United Kingdom, z.png@lncster.c.uk Sergei Svin The Whrton School, University of Pennsylvni, Phildelphi, PA 19104, svin@whrton.upenn.edu Tble A1 Nottion C = totl number of dily service slots A = number of dily service slots llocted for dvnce ppointments Z = the totl number of service slots mde vilble for dvnce ppointments on CB D 0 = sme-dy demnd λ 0 = expected vlue of the sme-dy demnd λ = expected vlue of dily dvnce ppointment requests ρ = λ/a, verge number of dvnce ppointment requests per service slot θ = frction of dedicted ptients X = number of ptients in the system L = length of ptient witing list L q = verge length of ptient witing list W q = verge ptient witing time M = trget witing time mesured in dys) r = reimbursement per ptient served l = dily penlty cost incurred by service provider for ech ptient on the witing list b = per-ptient diverting cost o = per-ptient overtime cost T = trnsfer pyment between the purchser nd the provider Π = expected profit for the provider Π p = expected cost for the purchser H - superscript representing provider with high overtime cost L - superscript representing provider with low overtime cost

2 Jing, Png nd Svin: Performnce-bsed Contrcts 2 Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 Appendix B: Proofs Proof of Proposition 1 Some preliminry results on stochstic dominnce for the modified M/D/1 queue re presented before we prove the results in Proposition 1. Let Xt) be the number of ptients in the ppointment queue t time t nd St) the residul service time for ptient in service if there is ny). Then the system cn be fully chrcterized by the two-dimensionl stte vrible Xt), St)). At ny point in time, the length of the queue is L q t) = Xt) 1) + nd the length of time n incoming ptient will wit is W q t) = Xt) 1) + + St). We pproximte the continuous-time system with discrete-time system with time intervls of equl length δ = 1/N, where N is lrge positive integer. Then, ech service time slot contins N successive intervls. The system stte cn be represented by x, s), where x is the number of ptients in the system nd s is the number of residul service time intervls of the ptient in service, s = 0, 1,..., N. Note tht we ssume tht s = 0 whenever x = 0. The Poisson rrivl process is pproximted by Bernoulli process such tht there is t most one ptient rriving in ech time intervl. We ssume tht the service strts t the beginning of time intervl nd occupies the entire time intervl. If there is ptient on the witing list t the beginning of service, his/her service strts immeditely fter the completion of the previous service. For instnce, if the system is empty t the beginning of n intervl, then the service of the first ptient to rrive during tht intervl strts t the beginning of the following intervl, t which point the system stte chnges from 0, 0) to 1, N). Assuming tht the system stte t the beginning of time intervl is x, 1), x > 0, nd there is no ptient rrivl during this this time intervl, the system stte chnges to x 1, N) t the end of this time intervl. If, however, there is ptient rrivl, the stte chnges to x, N). Note tht the stte x, 0) is observed only if x = 0. Similrly, if the system stte t the beginning of time intervl is x, i), x > 0, i > 1, the system stte chnges to x, i 1) if there is no ptient rrivl during tht time intervl, nd to x, i 1) otherwise. The two-dimensionl stte vrible cn be ggregted into single stte vrible i = Nx 1) + + s, which represents the totl number of time intervls n incoming ptient should wit to be served. Tht is, given ny stte i, the number of ptients in the system including the one in service) is i/n the nerest integer greter thn or equl to i/n) nd the number of time intervls of the residul service time of the ptient in service is i N i/n where i/n represents the nerest integer less thn or equl to i/n. Let ρ = λ/a. Given the policy prmeter, Z, the trnsition mtrix for the time-discretized Mrkov chin is represented by Π N = π i,j δ)), where 1 ρδ if i/n Z 1, j = i 1) +, ρδ if i/n Z 1, j = i 1) + + N, π i,j δ) = 1 θρδ if i/n > Z 1, j = i 1) +, θρδ if i/n > Z 1, j = i 1) + + N, 0 otherwise for i, j N. Let I represent the ggregted system stte t ny rndom time. Note tht we require ρθ < 1, which ensures the existence of the sttionry distribution of I. Denote the sttionry distribution by Q = [q 0, q 1,...], where q i is the sttionry probbility for the system stte i. Note tht Q = QΠ N = lim n QΠ n N. A sequence X = [x 0, x 1,...] is clled n incresing sequence if x i x i+1 for ny i = 0, 1,.... The next lemm shows tht mtrix Π N mps n incresing sequence into n incresing sequence.

3 Jing, Png nd Svin: Performnce-bsed Contrcts Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 3 Lemm B1. For ny incresing sequence X = [x 0, x 1,...], Y = Π N X is lso n incresing sequence. Proof. It is strightforwrd to see tht if i + 1)/N Z 1 or i 1)/N Z 1, then π i,i 1) +x i 1) + + π i,i 1) + +Nx i 1) + +N π i+1,i x i + π i+1,i+n x i+n, B1) where π i,i 1) + +N = π i+1,i+n nd is equl to ρδ or θρδ, nd π i,i 1) + = π i+1,i nd is equl to 1 ρδ or 1 θρδ. Otherwise, if i/n = Z 1, then [π i,i 1 x i 1 + π i,i 1+N x i 1+N ] [π i+1,i x i + π i+1,i+n x i+n ] = [1 ρδ)x i 1 + ρδx i 1+N ] [1 θρδ)x i + θρδx i+n ] = [1 ρδ)x i 1 + ρδx i 1+N ] [1 ρδ)x i + ρδx i+n + 1 θ)ρδx i x i+n )] 0, B2) where the lst inequlity follows from the fct tht X is n incresing sequence. Then, Π N X is lso n incresing sequence. Consider two prmeter triples, θ k, ρ k, Z k ), k = 1, 2. Let X k = [x k 1, x k 2,...], k = 1, 2, be two incresing sequences such tht X 1 X 2, i.e., x 1 i x 2 i for ll i. Let Π k N be the respective trnsition mtrices. The next lemm shows the monotone preservtion property of Π N. Lemm B2. If θ 1 θ 2, ρ 1 ρ 2 nd Z 1 Z 2, then Π 1 NX 1 Π 2 NX 2. Proof. We hve Π 1 X 1 = [π 1 0,0x π 1 0,Nx 1 N, π 1 1,0x π 1 1,Nx 1 N,..., π 1 NZ 1 1)+1,NZ 1 1)x 1 NZ 1 1) + π 1 NZ 1 +1,NZ 1 x 1 NZ 1,..., π 1 NZ 2 1)+1,NZ 2 1)x 1 NZ 2 1) + π 1 NZ 2 +1,NZ 2 x 1 NZ 2,...] [π 1 0,0x π 1 0,Nx 2 N, π 1 1,0x π 1 1,Nx 2 N,..., π 1 NZ 1 1)+1,NZ 1 1)x 2 NZ 1 1) + π 1 NZ 1 +1,NZ 1 x 2 NZ 1,..., π 1 NZ 2 1)+1,NZ 2 1)x 2 NZ 2 1) + π 1 NZ 2 +1,NZ 2 x 2 NZ 2,...] [π 2 0,0x π 2 0,Nx 2 N, π 2 1,0x π 2 1,Nx 2 N,..., π 2 NZ 1 1)+1,NZ 1 1)x 2 NZ 1 1) + π 2 NZ 1 +1,NZ 1 x 2 NZ 1,..., π 2 NZ 2 1)+1,NZ 2 1)x 2 NZ 2 1) + π 2 NZ 2 +1,NZ 2 x 2 NZ 2,...] = Π 2 X 2, where the first inequlity follows from the ssumption tht X 1 X 2 i.e., x 1 i x 2 i from the fct tht π 1 i,i 1) + π 2 i,i 1) + Let I k nd π 1 i,i 1) + +N π2 i,i 1), nd therefore + +N π 1 i,i 1) +x2 i 1) + + π1 i,i 1) + +N x2 i 1) + +N = x2 i 1) + + π1 i,i 1) + +N x2 i 1) + +N x2 i 1) +) for ll i) nd the second x 2 i 1) + + π2 i,i 1) + +N x2 i 1) + +N x2 i 1) +) = π2 i,i 1) +x2 i 1) + + π2 i,i 1) + +N x2 i 1) + +N. be the number of time intervls n incoming ptient will wit to be served corresponding to θ k, ρ k, Z k ), k = 1, 2. We sy tht I 1 is stochsticlly smller denoted by st ) thn I 2 if for ny incresing function h, E[hI 1 )] E[hI 2 )]. Let X k N be the number of ptients in the system corresponding to θ k, ρ k, Z k ). The next lemm proves tht the sttionry distributions of I nd X N re stochsticlly monotone in θ, ρ, nd Z. Lemm B3. If θ 1 θ 2, ρ 1 ρ 2 nd Z 1 Z 2, then I 1 st I 2 nd X 1 N st X 2 N. In ddition, W 1 q W 2 q, L 1 q L 2 q nd P rx 1 N Z 1 ) P rx 2 N Z 2 ).

4 Jing, Png nd Svin: Performnce-bsed Contrcts 4 Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 Proof. Let h ) : Z + R be ny incresing function. Then Y = [h0), h1),...] is n incresing sequence. Applying Lemm B2 yields Π 1 NY Π 2 NY. For ny integer n 1, we hve Π 1 N) n Y Π 2 N) n Y. Let Q k be the sttionry distribution of the number of ptients in the system corresponding to θ k, ρ k, Z k ). Then, Q k = Q k Π k N = lim Q 0 Π k N) n, n where Q 0 cn be ny strting distribution. So, we hve Q 1 Y = lim n Q 0 Π 1 N) n Y lim n Q 0 Π 2 N) n Y = Q 2 Y. Tht is, E[hI 1 )] E[hI 2 )] where I k, k = 1, 2 represent ggregted system sttes. Then, I 1 st I 2. Note tht the number of ptients in the system, X k N = I/N, is n incresing function of I k. Then, X 1 N st X 2 N. As W k q = E[I k δ], the stochstic monotonicity implies tht W 1 q W 2 q. Similr, s L k q = E[X k N 1) + ], then L 1 q L 2 q. The stochstic monotonicity lso implies tht P rx 1 N > 0) P rx 2 N > 0). Note tht under equilibrium the verge rrivl rte is ρ1 1 θ)p rx k N Z)) = ρ1 1 θ)p ri k NZ)). By the conservtion lw, P rx k N > 0) = ρ1 1 θ)p rx k N Z k )). Then, for ny θ < 1, P rx 1 N > 0) P rx 2 N > 0) implies tht P rx 1 N Z 1 ) P rx 2 N Z 2 ). As N δ 0), the Bernoulli process converges to the Poisson process nd the discrete-time system converges to the continuous-time system. Then the sttionry distribution of X N converges to the sttionry distribution of Xt), which, consequently, hs the stochsticlly monotone properties chrcterized in Lemm B3. Then, the monotone properties in Proposition 1 follow. Proof of Lemm 1 Proposition 1 shows tht W q A, Z)/A is monotone incresing in Z nd monotone decresing in A. Then for ny θ [0, 1], ny Z 0, nd ny A A, we hve W q A, Z) A W qa, ) A W qa, ) λ = A 2A A λ) = M, where the first equlity follows from 2). nd the second from the definition of A. This shows tht the service level constrint is stisfied. Proof of Proposition 2 ) Proposition 1 sttes tht P r XA t, Z t ) Z t ) is decresing in Z t for ny given A t, nd decresing in A t for ny given Z t. Then, the objective function of the first-best problem 13) is decresing in Z t, which implies tht the service level constrint must be stisfied s tightly s possible t the optiml solution, i.e., W q A t FB, Z t FB) /A t FB M while W q A t FB, Z t FB + 1) /A t FB > M. Now, consider the first-best solution T t FB, Z t FB, A t FB). Suppose tht Π t T t FB, A t FB, Z t FB) > 0. Then, since both T t nd Π t re monotone incresing in T t, we cn improve the objective function by lowering T t FB without violting the individul rtionlity constrint. Thus, Π t T t FB, A t FB, Z t FB) hs to be equl to 0. Then, 13) is obtined by replcing T t in the objective function of the purchser s problem by its expression from 6). b) As Proposition 1 sttes, W q A t, Z t ) /A t is incresing in Z t nd decresing in A t. This, in turn, implies tht Z t MA t ) is incresing in A t. Let A t F B B3) be the solution to 13). Observe tht the objective function in this problem is supermodulr in o t, A t ). Then, pplying Theorem 6.3 from Topkis 1978) Topkis, D.M Minimizing submodulr function on lttice. Opertions Reserch 262) ), we obtin tht A t F B is non-incresing in o t, which implies tht A H F B A L F B. Moreover, Z H F B = Z t A H F B) Z t A L F B) = Z L F B.

5 Jing, Png nd Svin: Performnce-bsed Contrcts Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 5 Similrly, the objective function is submodulr in b, A t ), nd, A t F B nd Z t F B re both non-decresing in b. Proof of Corollry 1 ) For o t = 0, 13) reflects the minimiztion of P r X A t, Z t ) Z t ). For ny finite Z t, s follows from Proposition 1, this objective is minimized by setting A t = C. Note tht for Z t +, the ppointment dynmics is identicl to one of the M/D/1 queue, nd lim Z t + P r X C, Z t ) Z t ) = 0, s long s the corresponding M/D/1 system is stble, i.e., s long s C > λ. This lst condition is implied by 2), which lso ensures tht the witing-time requirement is stisfied. b) For b = 0, the objective function in 13), for given Z t, is minimized by setting A t to the smllest possible vlue comptible with the service level constrint W A t, Z t )/A t. Since Proposition 1 shows tht W A t, Z t )/A t is n incresing function of Z t nd decresing function of A t, the vlue of Z t hs to be set t the lowest possible vlue. For Z t = 0, the ppointment dynmics becomes tht of n M/D/1 queue with Poisson rrivl rte of θλ, nd the ptient witing time constrint becomes θλ 2A t A t θλ) M At θλ 2 + θ2 λ 2 + θλ 4 2M. Thus, A t FB = θλ + θ 2 λ 2 + θλ M c) For θ = 1, the optimiztion objective is the sme s in prt b). At the sme time, θ = 1 lso implies tht the ppointment dynmics becomes tht of n M/D/1 queue with Poisson rrivl rte of λ, irrespective of the chosen vlue of Z t. Using the sme rguments, we obtin A t FB = λ + λ 2 + λ M Proof of Proposition 3 First, note tht, s the direct verifiction shows, the provider s objective function Π t r t, l t, A) is concve in A. The first-order optimlity condition for the provider of type t is: [ l t λ 2 1 A λ) A 2 ] o t 1 F D0 C A)) = l t λ2 2A λ) 2A 2 A λ) 2 ot 1 F D0 C A)) = 0. Under the contrct 15)-16), A defined in 4) stisfies the bove first-order optimlity condition s well s simple bounds constrints. The concvity of the objective function implies tht A is n optiml solution for the provider s problem. Finlly, it is esy to check tht given contrct 15)-16), the optiml solution A for the provider stisfies the service level constrint nd gives n objective function vlue o t E D0 [D 0 C +A ) + ], which is equl to the optiml objective function vlue for the purchser in the first-best solution. Hence, we hve proved tht the contrct 15)-16) chieves the first-best outcome. tht nd Proof of Proposition 4 ), b) First, for ny t < s, the incentive comptibility constrints 21) imply B4) B5) T t, A t, Z t ) Π ss T s, A s, Z s ) + o s o t )E D0 [D 0 C + A s ) + ], B6) Π ss T s, A s, Z s ) T t, A t, Z t ) + o t o s )E D0 [D 0 C + A t ) + ]. B7) Combining the bove two inequlities yields o s o t )E D0 [D 0 C + A t ) + ] T t, A t, Z t ) Π ss T s, A s, Z s ) o s o t )E D0 [D 0 C + A s ) + ], B8)

6 Jing, Png nd Svin: Performnce-bsed Contrcts 6 Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 which implies tht A t A s. Moreover, the bove inequlities lso imply tht Π 11 T 1, A 1, Z 1) Π 22 T 2, A 2, Z 2)... Π kk T k, A k, Z k). Then, employing the individul rtionlity constrints 20), we know tht the individul rtionlity constrints for the providers of type t < k re ll redundnt. Only type-k provider s individul rtionlity constrint needs to be verified. We clim tht Π kk T k, A k, Z k ) = 0 t the optimum. Otherwise, the purchser cn reduce the objective function vlue by decresing ll T t nd T k by the sme mount without violting the constrints 20) nd 21) until the constrint 20) is binding for t = k. Thus, the optiml trnsfer pyment designed for the type-k provider is equl to the sum of its overtime nd the ptient-diverting costs: T k = o k E D0 [D 0 C + A k ) + ] + bλ1 θ)p r XA k, Z k ) Z k). B9) We next clim tht, for ny t < k, the incentive comptibility constrint of type-t provider with s = t + 1 must be binding. Otherwise, from B6) nd B7), we cn observe tht the purchser cn reduce sme mount from ll T t, t < k, without violting other constrints, until B6) with s = t + 1 is binding. Thus, t the optimum, T t, A t, Z t ) = Π t+1,t+1 T t+1, A t+1, Z t+1) + o t+1 o t )E D0 [D 0 C + A t+1 ) + ]. B10) It is esy to exmine tht ll the other incentive comptibility constrints in 21) re redundnt. Combining the binding individul rtionlity constrint of type-k provider B9) nd the binding incentive comptibility constrints B10), we know tht for t = 1,..., k 1 which implies tht k 1 T t, A t, Z t ) = o τ+1 o τ )E D0 [D 0 C + A τ+1 ) + ], τ=t k 1 T t = [o t E D0 [D 0 C + A t ) + ] + bλ1 θ)p r XA t, Z t ) Z t )] + o τ+1 o τ )E D0 [D 0 C + A τ+1 ) + ]. The results of prts ) nd b) re obtined directly by substituting the expressions of T t obtined bove. c) In prt ), we hve shown tht A 1 SB A 2 SB... A k SB. Since ô t > o t for ll t > 1 nd ô 1 = o 1, the result of prt b) of Proposition 2 implies tht A t SB A t F B. Since the service-level constrints re binding t both first-best nd second-best solutions nd W q /A is incresing Z nd decresing in A, we hve the desired inequlities. Proof of Proposition 5 ), b) Clerly, Π ts r s, l s, A ts ) is strictly concve in A ts τ=t nd submodulr in o t, A ts ). Then, s follows from Theorem 6.3 in Topkis 1978) Topkis, D.M Minimizing submodulr function on lttice. Opertions Reserch 262) ), the mximizer of Π ts in o t, which implies tht A ts P A A ss P A nd A tt P A A st P A for t < s. in terms of A ts is decresing If λ < A ss PA < C, then A ss PA is n interior optimum of type-s provider s objective function, stisfying the first order condition λl s 2A ss PA λ) 2A ss PA Ass PA os 1 F D0 C A ss PA)) = 0. λ))2 B11)

7 Jing, Png nd Svin: Performnce-bsed Contrcts Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 7 Then, Πts Ats r s, l s, A ss ) = λls 2A ss PA λ) 2A ss PA Ass PA λ))2 ot 1 F D0 C A ss PA)) > 0. B12) The strict concvity of Π ts r s, l s, A ts ) with respect to A ts implies tht A ts PA > A ss PA. Similrly, if λ < A st PA < C, then A tt PA > A st PA. c) As λ/2a ts A ts λ)) is decresing in A ts, the provider s profit Π ts r s, l s, A ts ) is supermodulr in l s, A ts ). Then, s follows from Theorem 6.3 in Topkis 1978) Topkis, D.M Minimizing submodulr function on lttice. Opertions Reserch 262) ), A ts is incresing in l s. Proposition 3 sttes tht A is the optiml solution for the provider s optimiztion problem for l s = l t. Thus, A ts PA A if nd only if l s l t. In prticulr, from A < C nd Π ts r s, l s, A ts ) being strictly concve in A ts, it follows tht A ts > A for l s > l t. Proof of Proposition 6 ) nd b) As follows from Proposition 5, l t l t, since, otherwise, the type-t provider will choose A tt PA tht is less thn A, nd the witing-time constrint would be violted. Next, for ny t s, we hve PA) Π ts r s, l s, A ts PA) Π ts r s, l s, A ss PA) = Π ss r s, l s, A ss PA) + o s o t )E D0 [D 0 C + A ss where the first inequlity is from the constrint 28) nd the second is due to the optimlity of A ts PA nd the sub-optimlity of A tt PA). Then, for ny t < s, we hve o s o t )E D0 [D 0 C + A tt PA) + ] PA) Π ss r s, l s, A ss PA) o s o t )E D0 [D 0 C + A ss which implies tht A tt PA A ss PA. If A kk PA > A, then A tt PA A kk PA > A, which leds to the desired result. It suffices to show tht A tt PA > A for t < k if A kk PA = A. Note tht A kt PA C. If A kt PA = C, then it follows from Proposition 5 b) tht A tt PA A kt PA nd thus A tt PA = C > A. If A kt PA < C, then pplying Proposition 5 b) gin yields A tt PA > A kt PA, which implies tht Π kk PA) Π kt r t, l t, A kt PA) > Π kt PA) = PA) o k o t )E D0 [D 0 C + A tt where the first inequlity is from the constrint 28), the second strict) inequlity is by the optimlity of A kt PA, the strict concvity of Π kt r t, l t, A) in A nd the fct tht A kt PA < A tt PA. Then, the following inequlities hold: o k o t )E D0 [D 0 C + A tt PA) + ] > PA) Π kk which implies tht A tt PA > A kk PA = A. The desired result holds. PA) o k o t )E D0 [D 0 C + A kk c) Note tht in the trnsfer pyment the first term rλ 0 + λ) is linerly incresing in r nd is constnt to the provider. Then the chnges of r do not impct the providers cpcity lloction decision. Similr to the proof of Proposition 4, it is esy to show tht Π kk PA) = 0. Note tht A kk A. If A kk PA > A, then the preceding nlysis shows tht for ny t < k PA) PA) + o k o t )E D0 [D 0 C + A kk PA) + ] > o k o t )E D0 [D 0 C + A ) + ], B13)

8 Jing, Png nd Svin: Performnce-bsed Contrcts 8 Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 where the second inequlity is from the individul rtionlity constrint 27) nd the condition A kk PA > A. On the other hnd, if A kk PA = A, then by Proposition 5 ), A tk PA > A kk PA. Then, PA) Π tk r k, l k, A tk PA) > Π tk PA) = Π kk o k o t )E D0 [D 0 C + A ) + ], PA) + o k o t )E D0 [D 0 C + A kk PA) + ] where the first inequlity is from 28), the second strict) inequlity holds becuse Π tk r k, l k, A tk ) is strictly concve in A tk nd A tk PA is the unique optiml nd the fct tht A tk PA > A kk PA, nd the lst inequlity is due to the individul rtionlity constrint 27). Combining the bove two cses, the desired result holds. d) Recll tht for ny t < k we hve which implies tht Since A tt PA > A PA) Π kk PA) + o k o t )E D0 [D 0 C + A kk PA) + ] = o k o t )E D0 [D 0 C + A kk T t PA) o t E D0 [D 0 C + A tt PA) + ] + o k o t )E D0 [D 0 C + A kk PA) + ]. nd A kk PA A, we hve T t PA) > o k E D0 [D 0 C + A ) + ]. It is cler tht T k PA) o k E D0 [D 0 C + A kk PA) + ] o k E D0 [D 0 C + A ) + ]. Thus, the desired result holds. Proof of Proposition 7 If provider of type t chooses A t nd Z t tht violte the service-level constrint, then the definition of K implies tht the objective function vlue t A t, Z t ) for the provider is F t K o t E D0 [D 0 C + A t ) + ] bλ1 θ)p rxa t, Z t ) Z t ), which is smller thn o t E D0 [D 0 C + θλ) + ] o t E D0 [D 0 C + A t ) + ] bλ1 θ)p rxa t, Z t ) Z t ). The ltter is negtive number becuse A t θλ, which is necessry condition to ensure the stbility of the queueing system. On the other hnd, if provider of type t chooses A t = A t FB nd Z t = Z t FB, then the service-level constrint is not violted, the trnsfer pyment is F t, nd the objective function vlue t A t FB, Z t FB) for the provider is zero. Therefore, A t FB, Z t FB) is better solution thn ny solution A t, Z t ) tht violtes the service-level constrint for the provider of type t. Then, the problem for the provider of type-t is equivlent to mx F t o t E D0 [D 0 C + A t ) + ] bλ1 θ)p rxa t, Z t ) Z t ) ). A t,z t ) RM,C,θ,λ) Since F t is constnt, the provider s problem is further equivlent to min o t E D0 [D 0 C + A t ) + ] + bλ1 θ)p rxa t, Z t ) Z t ) ). A t,z t ) RM,C,θ,λ) By Proposition 2, the optiml solution for provider of type t is A t FB, Z t FB). Therefore, we hve proved tht the first-best outcome is chieved by the nonliner performnce-bsed contrct. Proof of Proposition 8 First, note tht for ny two types t > s, F t = o t E D0 [ D0 C + A t FB) +] + bλ1 θ)p rxa t FB, Z t FB) Z t FB) o s E D0 [ D0 C + A t FB) +] + bλ1 θ)p rxa t FB, Z t FB) Z t FB) o s E D0 [ D0 C + A s FB) +] + bλ1 θ)p rxa s FB, Z s FB) Z s FB) = F s.

9 Jing, Png nd Svin: Performnce-bsed Contrcts Article submitted to Mnufcturing & Service Opertions Mngement; mnuscript no. MSOM R2 9 Then, F 1 F 2... F k. Next we show tht for ny contrct F, K) stisfying conditions F F k nd F K o 1 E D0 [D 0 C + θλ) + ], provider of type t will lwys choose A t nd Z t such tht the service-level constrint is not violted. In fct, if provider of type t chooses A t nd Z t tht violte the service-level constrint, then the condition F K o 1 E D0 [D 0 C + θλ) + ] implies tht the objective function vlue for the provider is F t K o t E D0 [D 0 C + A t ) + ] bλ1 θ)p rxa t, Z t ) Z t ) o t E D0 [D 0 C + θλ) + ] o t E D0 [D 0 C + A t ) + ] bλ1 θ)p rxa t, Z t ) Z t ) 0, where the second inequlity is due to the stbility condition A t θλ. On the other hnd, if provider of type t chooses A t = A t FB nd Z t = Z t FB, then the service-level constrint is not violted, the trnsfer pyment is F t, nd the objective function vlue t A t FB, Z t FB) for the provider is F F t 0. Therefore, for provider of type t, A t FB, Z t FB) is better solution thn ny other solution A t, Z t ) tht violtes the service-level constrint. The problem for provider of type-t then becomes { mx F t o t E D0 [D 0 C + A t ) + ] bλ1 θ)p rxa t, Z t ) Z t ) }. A t,z t ) RM,C,θ,λ) Since F t is constnt, it is obvious tht the optiml solution for provider of type t is A t FB, Z t FB). Since the purchser s objective is to minimize her trnsfer pyment to the provider, the minimum trnsfer pyment is F. On the other hnd, the bove procedure hs shown tht F = F nd K = K is fesible solution to the purchser with trnsfer pyment of F. Furthermore, it is obvious tht ny contrct F, K) such tht F < F is infesible. Otherwise, the prticiption constrint of type-k provider will be violted i.e., Π kk < 0). Therefore, F, K) is n optiml threshold-penlty PBC contrct for the purchser. The results in prts b) nd c) of this proposition follow immeditely.