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2014 65 Gerardo Pelayo Rubio Analizando los efectos del diseño contractual y estructural en la cadena de suministro del sector salud Departamento Director/es Zaragoza Logistics Center Gürbüz, Mustafa Çagri Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Gerardo Pelayo Rubio ANALIZANDO LOS EFECTOS DEL DISEÑO CONTRACTUAL Y ESTRUCTURAL EN LA CADENA DE SUMINISTRO DEL SECTOR SALUD Director/es Zaragoza Logistics Center Gürbüz, Mustafa Çagri Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA UNIVERSIDAD DE ZARAGOZA TESIS DOCTORAL ANALIZANDO LOS EFECTOS DEL DISE ˜ NO CONTRACTUAL Y ESTRUCTURAL EN LA CADENA DE SUMINISTRO DEL SECTOR SALUD Gerardo Pelayo Rubio M´aster de Ingenier´ıa en Log´ıstica y Gesti´on de la Cadena de Suministro, Programa Internacional de Log´ıstica MIT-Zaragoza Zaragoza Logistics Center (ZLC), Universidad de Zaragoza (Espa˜na) Licenciado en Ingenier´ıa Industrial y de Sistemas, Instituto Tecnol´ogico y de Estudios Superiores de Monterrey (M´exico) 24 de Marzo de 2014 c Gerardo Pelayo Rubio. Reservados todos los derechos Autor: D. Gerardo Pelayo Rubio, Doctorando Director de tesis: Dr. Mustafa C¸agri G¨urb¨uz Profesor de Gesti´on de la Cadena de Suministro, Programa Internacional de Log´ıstica MIT-Zaragoza Zaragoza Logistics Center (ZLC) Director del Zaragoza Logistics Center (ZLC): Dr. David Gonsalvez Analizando los efectos del dise˜no contractual y estructural en la cadena de suministro del sector salud por Gerardo Pelayo Rubio en relaci´on con el cumplimiento parcial de los requisitos para la obtenci´on del t´ıtulo de Doctor en Log´ıstica y Gesti´on de las Cadenas de Suministro Resumen Introducci´on Balancear el acceso a medicamentos necesarios contra el aumento en los costes es uno de los retos fundamentales en el dise˜no y la reforma de los sistemas de salud. Del a˜no 2000 al 2008, el crecimiento promedio en el gasto per capita en productos farmac´euticos para los pa´ıses de la Organizaci´on para la Cooperaci´on y Desarrollo Econ´omico (OCDE) fue de casi 60 %. La problem´atica se encuentra especialmente presente en el proceso de introducci´on de nuevos medicamentos orientados al tratamiento de condiciones cr´onicas. Aqu´ı los precios de lista propuestos por los productores farmac´euticos tienden a ser altos para recuperar la inversi´on, en algunas ocasiones contrastando con la falta de evidencia robusta con respecto a la coste-efectividad del tratamiento al momento de la negociaci´on del precio de transferencia. M´as a´un, tal coste-efectividad puede variar a trav´es de las diferentes indicaciones terap´euticas de un mismo medicamento, i.e., para distintos grupos de pacientes. Como resultado, en los acuerdos tradicionales el pagador de salud (e.g., Sistemas Nacionales de Salud, Organizaciones de Mantenimiento de la Salud, grandes empresas aseguradoras) puede verse atrapado entre restringir el acceso al medicamento o arriesgar el pago de altos precios que pueden no justificarse ex-post debido a la incertidumbre sobre el valor real de la innovaci´on terap´eutica del medicamento, la falta de solidez en los resultados presentados por el productor, o la replicabilidad de esos resultados en la pr´actica cl´ınica. En respuesta a la creciente presi´on para controlar el gasto en el sector salud, los pagadores de salud han empujado a los productores farmac´euticos a reducir los precios, potencialmente reduciendo los incentivos para invertir en tratamientos innovadores, y continuamente resultando en la (temporal o definitiva) ausencia de un acuerdo entre ambos agentes involucrados con la consecuente p´erdida de bienestar para los pacientes potenciales y de beneficios financieros para el productor. Lo anterior ha motivado a los productores - particularmente aquellos en los sectores cardiovasculares y de oncolog´ıa - a explorar acuerdos m´as sofisticados donde los riesgos puedan ser compartidos de una manera m´as eficiente. Motivados por la tendencia mencionada, reconocemos que un pagador de salud debe decidir no ´unicamente si aprobar o no un nuevo medicamento para su (parcial o total) reembolso por consumo para la poblaci´on de pacientes que sirve, sino tambi´en determinar el nivel de servicio (cu´al ser´a el volumen adquirido para satisfacer la demanda de los pacientes), el nivel de acceso (cu´ales grupos de pacientes estar´an cubiertos por el pagador de salud), y las condicio5 nes de reembolso a los productores (los par´ametros del contrato). Adem´as reconocemos que un pagador de salud puede tener diferentes prioridades seg´un el ambiente social e industrial donde opere (e.g., maximizar la eficiencia de los recursos versus maximizar el bienestar de los pacientes), as´ı como restricciones (e.g., l´ımite m´aximo de gastos por periodo de demanda para alg´un medicamento o innovaci´on terape´utica, y un l´ımite m´ınimo de coste-efectividad). Con respecto a los productores farmac´euticos, consideramos que: la determinaci´on del precio de transferencia puede ser ex´ogena (a trav´es de precios de referencia externos) o end´ogena (a trav´es de acuerdos directos con los pagadores de salud); que pueden internalizar (parcial o totalmente) el riesgo de mantener el inventario; y que en algunos casos son capaces de segmentar el mercado a trav´es de la creaci´on de productos o canales distinguibles enfocados a cada grupo de pacientes. Preguntas de Investigaci´on En el contexto descrito donde un medicamento innovador con m´ultiples aplicaciones terap´euticas busca su introducci´on al mercado, la presente investigaci´on pretende responder de manera anal´ıtica las preguntas mostradas a continuaci´on. En un sistema verticalmente integrado, ¿c´omo interact´uan los niveles de acceso y de servicio en funci´on de las prioridades y restricciones del sistema? En una cadena de tipo productor - pagador de salud, ¿qu´e cambia cuando el precio es determinado de manera ex´ogena (vs. end´ogena) y el productor est´a (vs. no est´a) dispuesto a compartir los riesgos asociados a la incertidumbre en la magnitud de la demanda y en los resultados observados en los pacientes? Para un medicamento con m´ultiples aplicaciones terap´euticas, ¿c´omo se refleja la decisi´on de segmentar vs. consolidar el dise˜no/canal de distribuci´on, en el nivel de servicio y los incentivos para ejercer esfuerzo orientado a la innovaci´on? ¿Cu´al es el efecto de todo lo anterior en los beneficios del productor farmac´eutico, los gastos del pagador de salud, y el bienstar de los pacientes? De este modo, la investigaci´on espera contribuir a una comprensi´on m´as amplia del comportamiento del sistema, y as´ı eventualmente orientar el dise˜no de la estructura y los contratos en las cadenas de suministro del sector salud, de modo que exista una mejor alineaci´on con los objetivos de los agentes involucrados. Metodolog´ıa y Suposiciones Fundamentales El procedimiento general para responder a las preguntas anteriores se basa en una modelaci´on matem´atica de las situaciones previamente descritas utilizando la estructura del modelo del vendedor de peri´odicos (o newsvendor, como se le conoce normalmente en ingl´es). Esta elecci´on se debe a: i)los tiempos de espera extensos (aproximadamente 4 meses) para la construcci´on de capacidad productiva, aprovisionamiento de materias primas, producci´on, 6 Agradecimientos Durante estos a˜nos, he tenido la fortuna de cruzar caminos con una amplia variedad de personas, muchas de las cuales han dejado huella en mi desarrollo personal y profesional (una parte de lo cual est´a reflejado en estas p´aginas). A todos ustedes, mi gratitud y la esperanza de haber ofrecido algo valioso en reciprocidad. M´as all´a de esto, deseo dedicar un breve mensaje a un subconjunto de estas personas sin cuyo soporte probablemente no habr´ıa iniciado este camino, y seguramente no lo habr´ıa disfrutado como lo he hecho. A mi director de tesis, C¸agri, por su paciencia, su atenci´on al detalle, y su preocupaci´on por mi bienestar. A Santiago Kraiselburd y Prashant Yadav, con quienes las ra´ıces de la tesis fueron inicialmente desarrolladas, por ser mis primeros mentores y por continuar siendo una inspiraci´on. A Gast´on Cedillo, por su consejo, su apoyo incondicional, y por introducirme al mundo acad´emico. A Mozart Menezes, quien siempre tuvo fe. A Mario Monsreal, por todas sus lecciones y por creer en mi. A Florian Schick, por su amistad desinteresada. A mi familia, por estar siempre presente. Y a usted, lector, por tomarse el tiempo. ´ Indice general Resumen Ejecutivo 5 Agradecimientos 9 1. Introducci´on al Problema 19 1.1. Motivaci´on..................................... 19 1.2. Estructura de la Disertaci´on . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3. ElAn´alisis..................................... 25 1.3.1. Cap´ıtulo2................................. 26 1.3.2. Cap´ıtulo3................................. 27 1.3.3. Cap´ıtulo4................................. 27 2. Analizando el problema conjunto de acceso y cobertura en el sector salud 29 2.1. Introducci´on.................................... 29 2.2. MarcoTe´orico................................... 35 2.3. ElModelo..................................... 41 2.3.1. EscenarioGeneral............................. 41 2.3.2. El proceso de toma de decisiones . . . . . . . . . . . . . . . . . . . . 44 2.3.3. El canal verticalmente integrado . . . . . . . . . . . . . . . . . . . . 47 2.4. Extensiones para m´as de dos tipos de pacientes . . . . . . . . . . . . . . . . 76 2.4.1. Maximizando el bienestar esperado de pacientes dado I > 2 tipos de pacientes.................................. 77 2.4.2. Maximizando la funci´on esperada de utilidad del sistema dado que hay I > 2tiposdepacientes ......................... 81 15 2.5. Contratos de Precio-lineal Ex´ogeno . . . . . . . . . . . . . . . . . . . . . . . 84 2.6. Conclusiones.................................... 88 3. Analizando el valor de tres dise˜nos de contratos en el problema conjunto de acceso y cobertura en el sector salud 91 3.1. Introducci´on.................................... 91 3.2. MarcoTe´orico................................... 93 3.3. Contratos de Precio-lineal End´ogenos . . . . . . . . . . . . . . . . . . . . . . 98 3.3.1. Caso 1η: Maximizando el beneficio esperado de pacientes . . . . . . . 99 3.3.2. Caso 2η: Maximizando la funci´on de utilidad del pagador de salud . . 103 3.4. Contratos de precio ex´ogenos con capacidad de reabastecimiento . . . . . . . 106 3.4.1. Caso 1κ: Maximizando el beneficio esperado de pacientes . . . . . . . 111 3.4.2. Case 2κ: Maximizando la funci´on de utilidad del sector salud . . . . . 116 3.5. Contratos basados en desempe˜no . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5.1. El problema del pagador de salud . . . . . . . . . . . . . . . . . . . . 123 3.5.2. El problema del productor farmac´eutico . . . . . . . . . . . . . . . . 126 3.6. Conclusiones.................................... 127 4. Contratos en presencia de consumidores heterog´eneos: implicaciones del dise˜no estructural de la cadena en la innovaci´on, cobertura de pacientes, y beneficios 133 4.1. Introducci´on.................................... 133 4.2. MarcoTe´orico................................... 137 4.3. ElModelo..................................... 140 4.3.1. Escenario General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.3.2. Canales m´ultiples con integraci´on vertical . . . . . . . . . . . . . . . 143 4.3.3. Canal ´unico bajo integraci´on vertical . . . . . . . . . . . . . . . . . . 145 4.4. Contratos de precio-lineal ex´ogenos . . . . . . . . . . . . . . . . . . . . . . . 153 4.4.1. Canales m´ultiples con precio ex´ogeno (MX).............. 153 4.4.2. Canal ´unico con precio ex´ogeno (SX).................. 154 4.5. An´alisisNum´erico................................. 157 16 4.6. ExtensionesalModelo .............................. 164 4.6.1. Coordinando el dise˜no estructural de la cadena de suministro . . . . . 164 4.6.2. Funciones de coste dependientes del dise˜no estructural . . . . . . . . 166 4.6.3. Probabilidades de ´exito binarias . . . . . . . . . . . . . . . . . . . . . 167 4.7. El caso de dos categor´ıas de tipo estoc´astico . . . . . . . . . . . . . . . . . . 167 4.7.1. Canales m´ultiples con demandas estoc´asticas . . . . . . . . . . . . . . 168 4.7.2. Canal ´unico con demandas estoc´asticas . . . . . . . . . . . . . . . . . 169 4.8. Conclusiones.................................... 176 5. Comentarios Finales 181 Referencias 185 Ap´endice 1: Pruebas para el Cap´ıtulo 2 193 Ap´endice 2: Pruebas para el Cap´ıtulo 3 209 Ap´endice 3: Pruebas para el Cap´ıtulo 4 221 17 ´ Indice de figuras 1.1. Estructura de la disertaci´on . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2. Instant´anea de los Cap´ıtulos 2 y 3 . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3. Instant´anea del Cap´ıtulo 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1. Gasto farmac´eutico per capita (2000-2008) . . . . . . . . . . . . . . . . . . . 30 2.2. Crecimiento del gasto farmac´eutico per capita (2000-2008) . . . . . . . . . . 31 2.3. Calendariodeeventos............................... 44 2.4. Funciones esperadas de bienestar social y utilidad total, sin intersecci´on . . . 47 2.5. Funciones esperadas de bienestar social y utilidad total, con intersecci´on . . 48 2.6. Acercamiento a las funciones esperadas de bienestar social y utilidad total, conintersecci´on.................................. 48 2.7. Est´atica comparativa para q........................... 52 2.8. Orden de las cantidades de referencia (parte 1) . . . . . . . . . . . . . . . . . 58 2.9. Orden de las cantidades de referencia (parte 2) . . . . . . . . . . . . . . . . . 59 2.10. Toma de decisiones ´optimas bajo maximizaci´on de bienestar social (parte 1) 63 2.11. Toma de decisiones ´optimas bajo maximizaci´on de bienestar social (parte 2) 64 2.12. Encontrando ˜cen relaci´on al beneficio cl´ınico . . . . . . . . . . . . . . . . . . 67 2.13. Cambios en ˜cen relaci´on al valor de salvamento y al coste de demanda insatisfecha....................................... 68 2.14. Cambiando c(parte1) .............................. 70 2.15. Cambiando c(parte2) .............................. 71 2.16. Toma de decisiones ´optimas bajo maximizaci´on de utilidad neta (parte 1) . . 74 2.17. Toma de decisiones ´optimas bajo maximizaci´on de utilidad neta (parte 2) . . 75 2.18. Tres tipos de pacientes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 19 2.19. Toma de decisiones ´optimas bajo maximizaci´on de bienestar social con I= 3 80 2.20. Cambios en cpara I=3(parte1)........................ 82 2.21. Cambios en cpara I=3(parte2)........................ 82 3.1. Relacion entre wyQη τ(w)............................. 100 3.2. Efecto de wen los beneficios del productor bajo maximizaci´on del bienestar social, conforme Γ → ∞ ............................. 101 3.3. Efecto de wen los beneficios del productor bajo maximizaci´on de la utilidad neta del pagador de salud, conforme Γ → ∞ .................. 104 3.4. Secuencia de decisiones y eventos . . . . . . . . . . . . . . . . . . . . . . . . 120 4.1. Secuencia de decisiones y eventos . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2. Cambios en la cantidad ´optima a ordenar, dado que x=1 .......... 159 4.3. Cambios en la utilidad neta para diferentes valores de N, dado que x= 1 . . 160 4.4. Cambios a trav´es de βa.............................. 161 4.5. Cambios a trav´es de βb.............................. 162 4.6. Cambios en los esfuerzos relativos a trav´es de βb(ejemplo 1) . . . . . . . . . 163 4.7. Cambios en los esfuerzos relativos a trav´es de βb(ejemplo 2) . . . . . . . . . 163 20 Analyzing the effects of contract and structural design in health care supply chains A thesis presented by Gerardo Pelayo Rubio In partial fulfillment of the requirements for the degree of Doctor of Philosophy in Logistics and Supply Chain Management in the MIT-Zaragoza International Logistics Program at the Zaragoza Logistics Center, a research institute associated with the University of Zaragoza March 2014 c Gerardo Pelayo. All Rights Reserved. The author hereby grants to Zaragoza Logistics Center permission to reproduce and to distribute publicly printed and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. 1 [page left intentionally blank] 2 Acknowledgements Over the past five years, I’ve been fortunate to cross paths with a large number of individuals, many of whom have left a mark in my personal and professional development (and a part of which is reflected on these pages). To all of you, my gratitude and the hope of having had offered something valuable in return. Still, I wish to dedicate a few words to a subset of those individuals without whom I probably wouldn’t have started this path, and certainly wouldn’t have enjoyed it as much. To my advisor, C¸agri, for his unending patience, his attention to detail, and his sincere concerns for my personal and professional well-being. To Santiago and Prashant, with whom the roots of the dissertation was originally developed, for being my first mentors and for continuing to be an inspiration to this day. To Gaston, for his advice, his unconditional support, and for opening my eyes to the world of academia. To Mozart, who always had faith. To Mario, for all his lessons and for believing in me. To Florian, for his selfless friendship. To my family, who was always present. And to you, the reader, for taking the time. 9 [page left intentionally blank] 10 Author................................................... Gerardo Pelayo Rubio, PhD Candidate MIT-Zaragoza International Logistics Program ThesisSupervisor............................................. Dr. M. C¸agri G¨urb¨uz Professor of Supply Chain Management, Zaragoza Logistics Center Director,Ph.D.Program........................................ Dr. Maria Jesus Saenz Professor of Supply Chain Management, Zaragoza Logistics Center 11 [page left intentionally blank] 12 Contents Executive Summary 5 Acknowledgements 9 1 Introduction to the Problem 19 1.1 Motivation..................................... 19 1.2 Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3 TheAnalysis ................................... 25 1.3.1 Chapter2 ................................. 26 1.3.2 Chapter3 ................................. 27 1.3.3 Chapter4 ................................. 27 2 Analyzing the joint access and coverage problem in health care 29 2.1 Introduction.................................... 29 2.2 LiteratureReview................................. 35 2.3 TheModel..................................... 41 2.3.1 GeneralSetup............................... 41 2.3.2 The decision making process . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.3 The integrated channel . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.4 Extension for more than two types of patients . . . . . . . . . . . . . . . . . 76 2.4.1 Maximizing the expected social welfare given I > 2 types of patients . 77 13 2.4.2 Maximizing the system’s expected utility function given I > 2 types ofpatients................................. 81 2.5 Exogenous Price-only contracts . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.6 Conclusions .................................... 88 3 Analyzing the value of three endogenous contracting mechanisms in the joint access and coverage problem in health care 91 3.1 Introduction.................................... 91 3.2 LiteratureReview................................. 93 3.3 Endogenous price-only contracts . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.1 Case 1η: Maximizing expected social welfare . . . . . . . . . . . . . . 99 3.3.2 Case 2η: Maximizing Health’s expected value function . . . . . . . . . 103 3.4 Exogenous price contracts with capacity buffer allowed . . . . . . . . . . . . 106 3.4.1 Case 1κ: Maximizing expected social welfare . . . . . . . . . . . . . . 111 3.4.2 Case 2κ: Maximizing Health’s expected utility function . . . . . . . . 116 3.5 Performance-based contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 3.5.1 Health’sProblem............................. 123 3.5.2 Pharma’sProblem ............................ 126 3.6 Conclusions .................................... 127 4 Pulling, pooling, and contracting in the presence of heterogeneous consumers: implications of supply chain design on innovation, coverage and profits 133 4.1 Introduction.................................... 133 4.2 LiteratureReview................................. 137 4.3 TheModel..................................... 140 4.3.1 GeneralSet-up .............................. 140 4.3.2 Multiple channels under vertical integration . . . . . . . . . . . . . . 143 14 4.3.3 Single channel under vertical integration . . . . . . . . . . . . . . . . 145 4.4 Exogenous price-only contracts . . . . . . . . . . . . . . . . . . . . . . . . . 153 4.4.1 Multiple Channels with Exogenous Price (MX) ............ 153 4.4.2 Single Channel with Exogenous Price (SX) .............. 154 4.5 NumericalStudies................................. 157 4.6 ExtensionstotheModel ............................. 164 4.6.1 Coordinating the supply chain design . . . . . . . . . . . . . . . . . . 164 4.6.2 Design Dependent Cost Functions . . . . . . . . . . . . . . . . . . . . 166 4.6.3 Binary success probabilities . . . . . . . . . . . . . . . . . . . . . . . 167 4.7 When Pulling meets Pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.7.1 Multiple Channels with stochastic demands . . . . . . . . . . . . . . 168 4.7.2 Single channel with stochastic demands . . . . . . . . . . . . . . . . . 169 4.8 Conclusions .................................... 176 5 Final Comments 181 References 185 Appendix 1: Proofs for Chapter 2 193 Appendix 2: Proofs for Chapter 3 209 Appendix 3: Proofs for Chapter 4 221 15 [page left intentionally blank] 16 List of Figures 1.1 Structure of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.2 Snapshot of Chapters 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3 SnapshotofChapter4 .............................. 28 2.1 Pharmaceutical expenditure per capita (2000 - 2010) . . . . . . . . . . . . . 30 2.2 Pharmaceutical expenditure growth per capita (2000 - 2008) . . . . . . . . . 31 2.3 TimingofEvents ................................. 44 2.4 Expected social welfare and total utility functions with no intersection . . . 47 2.5 Expected social welfare and total utility functions with intersection . . . . . 48 2.6 Zoom on expected social welfare and total utility functions with intersection 48 2.7 Comparative statics for q............................. 52 2.8 Ordering of the reference quantities (part 1) . . . . . . . . . . . . . . . . . . 58 2.9 Ordering of the reference quantities (part 2) . . . . . . . . . . . . . . . . . . 59 2.10 Optimal decision making under expected social welfare maximization (part 1) 63 2.11 Optimal decision making under expected social welfare maximization (part 2) 64 2.12 Finding ˜cin relation to the health benefit . . . . . . . . . . . . . . . . . . . 67 2.13 Changes in ˜cin relation to goodwill costs and salvage value . . . . . . . . . . 68 2.14 Changing c(part1)................................ 70 2.15 Changing c(part2)................................ 71 2.16 Optimal decision making under system’s expected utility maximization (part 1) 74 2.17 Optimal decision making under system’s expected utility maximization (part 2) 75 17 2.18 Three Types of patients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.19 Optimal decision making under expected social welfare maximization for I= 3 80 2.20 Changes in cfor I=3(part1) ......................... 82 2.21 Changes in cfor I=3(part2) ......................... 82 3.1 Relationship between wand ¯ Qη τ(w)....................... 100 3.2 Effect of won Pharma’s profits under social welfare maximization, as Γ → ∞ 101 3.3 Effect of won Pharma’s profits under Health’s net utility maximization, as Γ→ ∞ ....................................... 104 3.4 Sequence of decisions and events . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.1 Sequence of decisions and events . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2 Changes in the optimal order quantity, given x=1 .............. 159 4.3 Changes in optimal net utility for different values of N, given x= 1 . . . . . 160 4.4 Changes across βa................................. 161 4.5 Changes across βb................................. 162 4.6 Changes in the relative effort levels across βb(example 1) . . . . . . . . . . . 163 4.7 Changes in the relative effort levels across βb(example 2) . . . . . . . . . . . 163 18 innovation efforts are modeled. The results, which hope to orient public-policy making, are provided in terms of firm profitability, health care spending, drug access, and patient welfare. Methodologically, the research contributes firstly to the operations literature by translating familiar concepts such as the price-dependent newsvendor and the pooling effect in a new setting with particular characteristics for which results are not fully explained in previous works; and secondly, to the health economics literature by simultaneously modeling demand and outcome uncertainty. The detailed discussion of the theoretical contributions and conclusions derived from the analysis is provided within each chapter. Figure 1.1: Structure of the dissertation 1.3 The Analysis Chapter 1 provides an extended introduction to the problem. Chapters 2 and 3 focus on the simultaneous access and service level decisions, while Chapter 4 takes the access level as exogenously given and analyzes the optimal structural design and effort decisions. Table 2 and Table 3 provide a snapshot of the analysis, along with the key associated assumptions. 25 1.3.1 Chapter 2 We begin the analysis by modeling the introduction process of a new drug treatment that can be used by multiple patient categories who benefit differently from it. A profit-maximizing pharmaceutical manufacturer offers to sell the new drug to a health-payer, who decides the access and service levels for the patient population he serves. An analytical comparison is done assuming that the health-payer either maximizes patient welfare, or maximizes the entire utility function (i.e., incorporating purchasing costs). Under both decision-making criteria two constraints are included: an absolute budget constraint to set a limit on health care spending, and a cost-effectiveness constraint to maintain a balance between costs and beneFigure 1.2: Snapshot of Chapters 2 and 3 26 fits. First, the analysis for the vertically integrated chain is presented both as an efficiency benchmark and as a simplified setting for understanding the dynamics between access and service level under the problem’s particular characteristics. Second, the exogenous price contract is formulated, setting the grounds for the analysis of contracts where the manufacturer can endogenously determine at least some contract parameters. 1.3.2 Chapter 3 In this chapter we keep the structure of the model presented earlier but focus on the decisions made by the pharmaceutical manufacturer, given that she can anticipate the healthpayer’s access and service level decisions. Specifically, we analyze three contracting mechanisms: endogenous price-only contracts; exogenous price contracts with capacity buffer; and performance-based contracts. Endogenous price-only contracts have been thoroughly studied, but not in a supply chain setting as the one we consider where the downstream player has such decision space (as mentioned above for Chapter 2). The last two contracts are novel proposals based on existing models, but adapted to the needs of the system. The virtues and drawbacks of each mechanism are detected, with an emphasis on the search for Pareto improvements. 1.3.3 Chapter 4 The last part of the dissertation departs from the analysis of the access level decision and captures a different consequence of patient heterogeneity by analytically comparing the performance of two supply chain designs. Under the first design, (up to) two patient categories are served by a single inventory stock on a first-come first-serve basis, while on the second design a dedicated inventory stock is used to serve each patient category. It is assumed that the realization of the second category is stochastically contingent on innovation efforts made by the pharmaceutical manufacturer, that such manufacturer chooses the supply chain 27 Figure 1.3: Snapshot of Chapter 4 design by having the option to commercialize two differentiated products (e.g., through different presentations, different delivery formats, exclusive distribution channels), and that the health-payer is responsible for making the inventory decision. First, the optimal decision path for a vertically integrated chain is analyzed, and then the incentive misalignments derived from vertical separation are explained along with some theoretical and managerial extensions to the model. 28 Chapter 2 Analyzing the joint access and coverage problem in health care 2.1 Introduction A report in the United Kingdom (UK) by the Rarer Cancers Foundation shows that since 2009, 18 new treatments were rejected by the National Institute of Clinical Excellence (NICE) - many because they were not deemed “cost-effective” - out of 34 put forward.1In Australia, the federal government announced on February 2011 that to return the budget to a surplus, no new drugs would be added to the Pharmaceutical Benefits Scheme (PBS) until 2013, irrespective of the recommendations of the Pharmaceutical Benefits Advisory Committee (PBAC).2Balancing access to needed medicines against escalating costs is one of the most challenging tasks in health care reform (Chalkidou, Lopert and Gerber, 2012). From 2000 to 2008, the average growth in the per capita spending on pharmaceuticals for Organisation for Economic Co-operation and Development (OECD) countries was almost 60% (and exceeding 70% by 2010, considering the subset of countries for which information is available); Figures 1http://www.dailymail.co.uk/health/article-2194998/Number-cancer-drugs-rejected-health-watchdogrises-50-years.html 2Following criticism by pharmaceutical industry, providers, and patients, the measure was removed on September the same year. 29 Figure 2.1: Pharmaceutical expenditure per capita (2000 - 2010) Note: Expenditures are expressed in U.S. dollar purchasing power parity. Source: 2012 OECD Health Data, http://www.oecd.org/health/healthpoliciesanddata/oecdhealthdata2012-frequentlyrequesteddata.htm 2.1 and 2.2 provide more specific information about some of the largest/most influential markets. The trade-off is particularly present in the introduction of new drugs aimed at treating chronic conditions where list prices proposed by the pharmaceutical manufacturers tend to be high in order to recoup their investment, sometimes contrasting with a lack of robust evidence regarding the cost-effectiveness of the treatment at the time when price is negotiated; moreover, such cost-effectiveness may vary across a drug’s different indications, i.e., for different patient groups. As a result, in traditional agreements a health-payer - e.g., National Health Systems, Health Maintenance Organizations, large insurance companies - may be forced either to restrict access or to risk paying high prices that are not ex-post justified due to the uncertainty about the real value of a drug’s therapeutic innovation, the lack of solidity of the results presented by the manufacturer, or the replicability of those results in clinical practice. But as pressures to control health care spending keep increasing, healthpayers have pushed pharmaceutical manufacturers to decrease prices, potentially decreasing the incentives to invest in innovative treatments, and often resulting in the (temporary or definitive) absence of an agreement between both players at the loss of patient welfare and manufacturer’s profits. The United Kingdom’s Department of Health has been one of the most innovative players regarding the relationship between the pharmaceutical manufacturers and health-payers. 30 Figure 2.2: Pharmaceutical expenditure growth per capita (2000 - 2008) Source: 2012 OECD Health Data, http://www.oecd.org/health/healthpoliciesanddata/oecdhealthdata2012-frequentlyrequesteddata.htm The Pharmaceutical Price and Regulation Schemes (PPRS) - a non-contractual agreement renegotiated every five years between the UK Department of Health and the Association of the British Pharmaceutical Industry (ABPI) - were first introduced in 1957 as a mechanism to ensure access to good quality branded medicines at reasonable prices to the National Health Service (NHS) and fair returns to the pharmaceutical industry, where the price is regulated by setting profit caps for pharmaceuticals. In 1999, NICE was established, playing a key advisory role in technology appraisal by quantifying benefits in a consistent and comparable way across the full range of health-related conditions and applying an Incremental Cost-effectiveness Ratio (ICER) methodology; it is important to note that NICE’s recommendation is by law a sufficient but not necessary condition for inclusion of the drug in the NHS list. In the existing system based on the PPRS 2009, a standard willingness-to-pay (WTP) threshold is applied to all new products so that a drug will be recommended for inclusion in the NHS if the cost per QALY3achieved (i.e., the incremental cost-effectiveness ratio) is less than £20,000, and analyzed on a case by case basis when the ICER is between £20,000 and £30,000 (i.e., the WTP threshold). However, the recurrent rejection of new treatments, often on the grounds of lack of cost-effectiveness evidence, has motivated pharmaceutical manufacturers - particularly in the cardiovascular and oncology sectors - to explore more sophisticated agreements where risks can be more efficiently shared. Pressured by public and industry lobbying, and following the recommendations from the market 3NICE has used the Quality Adjusted Life Years (QALYs) - a QALY is the amount of health represented by a year of life at full health - to measure the benefits of an intervention. 31 study performed by the Office of Fair Trading (OFT, 2007), the NHS is expected to make a transition towards a value-based system of pricing medicines. Two temporary solutions are currently in practice: the Cancer Drug Fund which provides £200 million per year to fund cancer drug treatments not recommended by NICE that physicians deem appropriate for a particular patient; and Patient Access Schemes where manufacturer and payer agree on an evaluation time, a verifiable measurement, and a target, so that the manufacturer offers a discount or rebate to the NHS when the target is not reached. However, both programs are expected to be substituted in 2014 by value based pricing (VBP) which no longer considers a unique threshold across all interventions, but rather has a base threshold which is explicitly increased for high burden of illness, therapeutic innovation, and wider societal benefits, i.e., the threshold may be different for different drugs and/or indications. The approach provides manufacturers with freedom to propose prices as long as the threshold is satisfied, and expects to increase transparency in the technology appraisal process and predictability for pharmaceutical manufacturers. Other countries are following the trend of implementing mechanisms for using comparative clinical and cost-effectiveness to inform technology adoption and regulate selling prices. In Australia, the PBAC is responsible for recommending inclusion of drugs in the national formulary for reimbursement, using a variable cost-effectiveness threshold contingent on each drug’s characteristics. As opposed to the UK, in Australia the PBAC recommendation is a necessary but not sufficient condition for the Minister for Health and Ageing to approve a drug’s inclusion. Risk sharing contracts have included the use of rebates paid to the government when expenditures exceed an annual cap, a pooled annual sales cap for a group of drugs that treat a common condition, and price-volume agreements. The Life Saving Drugs Program operating outside the PBS has been set up to provide free access to certain expensive, life-saving drugs for rare, serious, life-threatening conditions, currently funding 8 drugs for almost 200 patients. Germany uses reference pricing groups to regulate prices 32 of drugs. Following the establishment in 2004 of the Institute for Quality and Efficiency in Health Care (IQWiG) - modeled after NICE -, since January 2011 reference pricing is only used for drugs that do not demonstrate additional benefits; for those that do, an agreement between the manufacturer and the national association of statutory health funds (SHI) must be reached within 6 months or else a central board of arbitration determines a rebate based on international prices. In the United States, the establishment of the Patient-Centered Outcomes Research Institute recognizes the relevance of evidence-based decision-making, and Medicare’s Independent Payment Advisory Board is expected to motivate the use of price-negotiations and risk-sharing agreements. Some detailed, recent surveys on the use of risk sharing contracts across different countries include Pugatch, Healy and Chu (2010), and Espin, Rovira and Garcia (2011). However, as such papers comment, many of the so-called risk-sharing agreements seem to act as cost-containment mechanisms rather than a true reassignment of risks and benefits to both players. Motivated by the above trend, we understand that a health-payer must decide not only whether to accept a new drug under (partial or full) reimbursement for the patient population it serves, but also determine the volume purchased (how many patients are expected to be treated), access level (which patient groups will be serviced by the health-payer), and reimbursement conditions to the manufacturers (contract parameters). In our model, we explicitly acknowledge that each health-payer may have different priorities affected by the social and industry environment where it operates, and that the manufacturer may hold an information advantage about a drug’s expected future value. Moreover, a health-payer’s decision may be limited by absolute and relative expenditure restrictions. To exemplify the former, a prostate cancer drug that privately costs around £3,000 for a month’s supply had been offered to the NHS at a discount, but NICE declared the number of men who need the drug would make it financially unworkable.4As for the latter, the ICER methodology which 4Retrieved Oct. 8, 2012 from http://www.savistamagazine.com/news/prostate-cancer-drugprovisionally-rejected-by-nhs 33 lies at the heart of any decision incorporating comparative clinical and cost-effectiveness, requires an upper bound or threshold that essentially makes the maximum allowable level of expenditures a function of the benefits. As a result, the scope of this chapter is to analyze: a) the change in the system’s optimal decisions as a function of the health-payer’s decision-making priority, constraints, and the contract parameters using the newsvendor framework; b) the way in which double marginalization and asymmetric beliefs between the pharmaceutical manufacturer and the healthpayer influence the mechanics; and c) the impact of all the latter on manufacturer’s profits, health-payer’s costs, and patient access and service levels. Our contribution can be summarized in three parts. First, we derive an efficient algorithm for determining the optimal access and service level policy, based on two very easy to calculate thresholds, in a setting based on the price and quantity newsvendor model but where the feasible prices are not continuous5, the decision space is constrained, and the decision maker’s priorities may vary. And second, we comment extensively on the situations - as a function of the combination of parameters - that are likely to induce full or restricted access, providing interesting insights for policy makers and pharmaceutical manufacturers. The rest of the chapter is introduced with a literature review of relevant work. Section 2.3 introduces the model and solves the problem for a single decision maker under two types of patients. Section 2.4 expands the results to nnumber of patient types, and derives an algorithm to efficiently find the optimal solution without the need of full enumeration. When the pharmaceutical manufacturer and the health-payer act separately, section 2.5 expands the results to incorporate exogenous price-only contracts, setting the base model for the next Chapter. Conclusions and further research opportunities are discussed in §2.6. 5In our model, the additive part of the objective function is given by the health benefits obtained by those patients who receive the drug under analysis. As will be thoroughly explained in §2.3, there is a direct mapping between the access level chosen, and the average expected health benefits received by the patients, i.e., the retail price in typical newsvendor models. 34 we take a supply chain approach rather than focusing on a single echelon, by explicitly considering the role (and in Chapter 3, the best response function) of the manufacturer. Second, the “expected price”, i.e., the health benefit, is assumed to be discrete in our model, which complicates the analysis and requires a different treatment. Third, in addition to expanding the analysis for the traditional objective function in the price dependent newsvendor model, we consider a second alternative (maximizing social welfare, which will be formally defined below) motivated by the mission of some of the institutions, usually public, in charge of finding the optimal values for the model’s decision variables. And fourth, we incorporate an absolute and a relative budget contraints, simultaneously, which has relevant implications on the feasible decision space, and therefore, on the solution process to find the optimal solution. In sum, our approach intends to aid public policy making and firm strategy by analytically showing the changing supply chain impact of different decision-making criteria by the payer under two fundamental constraints driven by the growing pressures surrounding health care costs, and in particular, pharmaceutical spending. 2.3 The Model 2.3.1 General Setup Consider a health care supply chain where a risk-neutral pharmaceutical manufacturer, hereafter Pharma and denoted by subindex m, offers to sell a new (prescription) drug to a riskneutral6central health care system (i.e., the health-payer and the health-provider are part of the same governing institution), hereafter Health and denoted by subindex h, through some take-it or leave-it contract agreement. Let Nbe a random variable representing the number 6This assumption, while inconsistent with the current observations regarding the introduction of new drugs aimed at treating chronic conditions will be relaxed in Chapter 3. As will be explained there, its relaxation under price-only contracts provides no additional insights and is therefore removed in the present chapter to avoid unnecessary noise. 41 of patients that arrive to receive treatment through Health within a single finite time period, where Nfollows a Poisson distribution with parameter λ. We assume heterogeneity within patients to exist so that upon arrival to Health, patients are categorized into 2 mutually exclusive types based on their physiological conditions and medical history.7Let f(i) be the probability that an incoming patient is categorized as type i, i = 1,2, and F(i) be the probability that a patient is categorized as type ior lower, e.g., F(2) = f(1) + f(2), with F(0) = 0, F(2) = 1. Henceforth we assume f(1) = θ, and f(2) = (1 −θ). Let birepresent the multiplication of the incremental value of the health gains received by a patient categorized as type i,i= 1,2,, and the health-payer’s ceiling ratio; i.e., biis the incremental value derived from the drug’s administration times the health-payer’s maximum willingness-to-pay for that value. For simplicity of exposition, we will henceforth refer to bisimply as health benefits. Without loss of generality, we organize patient categories such that b1> b2. This implies that the perceived value to the health-payer from a drug being administered to a patient of category 1 versus to a patient of category 2, is b1−b2>0; this can occur based on (a)superior clinical outcomes in patients of category 1; (b)on a higher ceiling ratio for category 1 originated by the drug’s larger societal benefits, the disease severity, or the degree of innovation for that particular patient category; or (c)on a combination of both. The formulation therefore allows for a dynamic cost-effectiveness threshold, e.g., it allows for the existence of different willingness-to-pay thresholds for each patient category, as is already the case in Australia and will be in the UK starting 2014. We approximate the trading opportunities between Pharma and Health using the single period newsvendor framework where Pharma must commit production well in advance of receiving Health’s order. This is based on (a)the long lead times in building production capacity, sourcing raw materials, manufacturing the drug and delivering it to Health; (b)the 7We assume that all incoming patients are diagnosed in order to learn their medical status, and as a result even in a fully inclusive policy, the examination cost is a constant which is not explicitly included in the model. 42 quantity discounts offered for large purchases which makes the partition of demand into long periods a reasonable assumption; and (c)the manufacturer’s high utilization levels, given limited manufacturing capacity and possibly multiple clients, which reduces her ability to satisfy larger than expected demand in the short term.8Additionally, we consider that the newsvendor framework can be useful in representing the collateral costs associated with demand forecast mismatches exacerbated by the growing budget pressures in the health care sector. On one hand, overestimating demand may result in a portion of the health care budget being “trapped” in anticipation for more patients, possibly rejecting or delaying inclusion of other treatments into the formulary listings. On the other hand, underestimating demand may result in higher than expected costs, either in terms of health outcomes because patients are not treated in a timely manner with the best available treatment, or in monetary terms because the treatment is available but creates a budget deficit that negatively affects future introduction of innovative treatments. The order of events in the model is depicted in Figure 2.3 and described next. (1)Pharma announces bifor i= 1,2, and announces the selling price for the drug, which may be either exogenously or endogenously determined as will be explained in the coming sections. (2)Health selects an order quantity Qof drugs and a prescription policy threshold τso that an incoming patient of type iis prescribed the drug only if i≤τ; this means that the probability that an incoming patient is an eligible candidate for receiving the drug is F(τ). Mathematically, the conditional demand for the drug given some prescription policy threshold can be seen as a random number Nof trials, each with a success probability F(τ); by the Poisson property, the effective demand,D(λ, τ), is also Poisson distributed with parameter λF(τ). Let p(x;λF(τ)) = (λF(τ))x x!e−(λF(τ)), be the probability that exactly xpatient arrivals from the effective demand occur during the period; and let P(x;λF(τ)) = ∞ X j=x p(j;λF(τ)) be the complement of the Poisson CDF. We can now define B(τ),Pτ i=1 bif(i) F(τ)to be the 8This assumption is relaxed in Chapter 3 through the introduction of a capacity buffer contract. 43 expected health benefits obtained by a patient belonging to the effective demand that is eligible to receive the drug treatment. (3)Pharma produces and delivers to Health Qunits of the drug at marginal cost c. (4)Demand is realized. Excess drugs may be salvaged at a per unit value δ, which may be interpreted either as the opportunity cost or as a discounted sale to a secondary market; to avoid trivial problems, assume δ<c<b1. If D(λ, F(τ)) > Q, a per unit cost, g, is accrued to Health for each patient arrival that satisfies the prescription policy threshold but does not receive the drug treatment due to a stock-out. To keep integrality, we will use bxcand dxeas the floor and ceiling functions, respectively. For the moment, all players are assumed to hold symmetric information about all functional forms and parameters. Figure 2.3: Timing of Events 2.3.2 The decision making process Define A(Q, τ), E [min[Q , D(λ, τ)]] to be the expected quantity of administered drug treatments; E [max[0 , Q−D(λ, τ)]] = (Q−A(Q, τ)), to be the expected leftovers for Health; and E [max[0 , D(λ, τ)−Q]] = λF(τ)−A(Q, τ), to be the expected quantity of understocked units of the drug at the end of the period. Finally, let T(·) be the payment from Health to Pharma as a function of the contract parameters; it is well-known that when Pharma and Health act as a single decision-maker, it is optimal to set a transfer payment T=cQ to prevent double-marginalization. Then, for a risk-neutral single decision-maker the social 44 welfare expected utility function is: S(Q, τ) = B(τ)A(Q, τ) + δ(Q−A(Q, τ)) −g(λF(τ)−A(Q, τ)) ,(2.3.1) and the system’s expected utility function is: Z(Q, τ) = −cQ +B(τ)A(Q, τ) + δ(Q−A(Q, τ)) −g(λF(τ)−A(Q, τ)) =−cQ +S(Q, τ) (2.3.2) Based on the latter, in order to better understand the impact of the different priorities used in practice by health care systems, we explicitly distinguish between two possible criteria for the decision-making process: •Maximize utility of social welfare, S(Q, τ); this is the case when Health’s priority are the recipients of the drug treatments, i.e., the patients, and we consider it a more appropriate approach in settings where the health-payer is a non-for-profit institution as occurs in several national health systems. For example, in Germany a 2005 Court Decree establishes that treatment in the case of a life-threatening disease is an essential part of health care and statutory health funds must pay for it. •Maximize the decision maker’s utility function, Z(Q, τ); this is the case when the decision maker’s priority is to make an efficient use of its resources and we consider it a more appropriate approach in settings where the health-payer’s priority is to maximize the use of its resources or when the payer is a for-profit institution, e.g., private insurance companies, as occurs in a large portion of the the United States market and some developing countries. For example, in Australia the PBAC recommendation may be accompanied by closely specified access restrictions due to a drug’s lack of robust clinical evidence of a clinically important additional benefit, or because the incremental costs of obtaining those benefits mean that drugs are cost-effective in only a defined 45 group of patients. In other words, a drug’s high cost-effectiveness for one of the indications cannot be considered to subsidize its use for an indication which benefits are on its own not cost-effective. In addition, the reality is that for both decision-making criteria, the amount of resources is limited and the health-payer’s decision space is typically bounded by a set of minimum conditions that should be satisfied in order for a drug treatment to be approved for a particular segment of the patient population. In response to these issues, two constraints are included in our analysis: a budget constraint: T(·)≤Γ, where Γ is an exogenous upper limit on Health’s expenses for the drug under analysis; and a cost-effectiveness constraint: Z(Q, τ)≥0, which makes sure that the expected net benefits derived from the drug’s approval are above some minimum threshold. This approach to evaluating new technologies is known as net monetary benefits (NMBs), and as long as the terms in the calculations of incremental benefits and costs are the same, then positive NMBs are equivalent to the ICER being less than the willingness to pay in a cost-effectiveness analysis. Next, we solve both the situations when an integrated decision-maker: maximizes expected social welfare utility (§3.3.1), and maximizes the system’s expected utility function (§3.3.2). 46 2.3.3 The integrated channel We refer to this setting with the symbol ς, where ςis used to denote the single decision-maker structure. Before going further, some useful structural properties of the model are presented. Lemma 1: For a given τ, the social welfare function, S(Q, τ), is increasing and concave in Q. Lemma 2: For a given τ, the system’s utility function, Z(Q, τ), is concave in Q. Figures 2.4, 2.5, and 2.6 provide a graphical representation of Lemmas 1 and 2, and are useful in explaining the intuition created by our formulation. For increasing values of the order quantity, Q, the solid lines in Figures 2.4 and 2.5 plot the expected social welfare curves while the dashed lines plot the expected system’s utility curves. Notice first that from the formulation, when the access level is restricted, the value of b2is irrelevant as only high health benefit patients are treated. Second, the reason for the steeper slope in the social welfare functions in Figure 2.4 versus 2.5 after reaching the maximum point in the expected utility functions is driven by the higher value of δin Figure 2.4; this happens because as the incremental number of patients who are administered the drug goes to zero, the social welfare curve increases at a constant rate δ. The obvious consequence is that as the salvage Figure 2.4: Expected social welfare and total utility functions with no intersection λ= 600; θ= 0.4; b1= 1; b2= 0.15; g= 0; δ= 0.2; c= 0.3 47 Figure 2.5: Expected social welfare and total utility functions with intersection λ= 600; θ= 0.4; b1= 1; b2= 0.5; g= 0.2; δ= 0.05; c= 0.3 Figure 2.6: Zoom on expected social welfare and total utility functions with intersection λ= 600; θ= 0.4; b1= 1; b2= 0.5; g= 0.2; δ= 0.05; c= 0.3 value is positive, the unconstrained social welfare maximizer will continue to order indefinitely. Third, when g= 0 both access levels provide a utility of zero when no drugs are ordered, while when g > 0, the full access policy is inferior to the restricted access policy when Q= 0. This occurs because under full access, more patients are expected to arrive, resulting in higher costs of understocking. Fourth, under restricted access, the curves initially grow at a faster rate due to the higher value of the average health benefits, but they do so for a shorter range of order quantities due to the decreasing probability that demand exceeds a particular order quantity. These effects are generated by the value of θand the shape of the demand distribution. As θincreases, the slope of the expected utility under the restricted access policy will stay positive for higher values of Q; and also the initial increasing slope for both access level policies will get closer to each other as θgets closer to 1. Finally, it is worth noting that in Figure 2.5 the access level policies cross paths, while in Figure 2.4 they do not. Figure 2.6 zooms in on the crossing point for the same parameter combination used 48 in Figure 2.5; since due to integrality there may not exist an integer order quantity where the functions are equal, henceforth when we speak of a crossing point or an intersection, we refer to a change in dominace of one access level curve versus the other. Proposition 1 provides the conditions for this crossing to occur, and explains the changes in the crossing point as a function of the problem’s parameters. Proposition 1: Let qbe a positive order quantity such that S(q, 1) ≥S(q, 2); and S(q+1,2) > S(q+ 1,1). a) If inequality (2.3.3) is satisfied, then qis unique and given by equation (2.3.4). θ≥1−P(Q;λ) 1−P(Q;λθ)P(Q+ 1; λθ) P(Q+ 1; λ)(2.3.3) q=max Qg b1−δ+g λ A(Q, 2)+b1−b2 b1−δ+g≥A(Q, 2) −A(Q, 1) A(Q, 2)  1 1−θ (2.3.4) b) b2> δ is a necessary and sufficient condition for qto exist. c) If b2< δ, then S(Q, 1) > S(Q, 2),∀Q > 0. d) If b2=δ, then S(Q, 1) > S(Q, 2), for some finite Q > 0, and limP(Q,λ)→0S(Q, 1) − S(Q, 2) = 0. e) The value of qis increasing in b1, g, δ, and decreasing in b2. The change in qwith respect to θis ambiguous. The existence of a unique qis critical in our analysis because it provides a threshold value for the dominance of either of the access level policies, both in terms of maximizing expected social welfare or expected system’s utility - because the manufacturing cost is linear in the order quantity. In other words, qis independent of the transfer price and is then the same for the social welfare curves and for the system’s utility curves. It is useful in representing the 49 essential tradeoff between choosing higher access levels versus higher service levels; notice that for any fixed order quantity, the service level will always be higher for lower levels of access. Consequently, qwill be utilized below as a reference point to determine the optimal policy given the objective function and cost-effectiveness and budget constraints. Proposition 1a gives the conditions for finding qequation (2.3.4) -, and for such qto be unique - equation (2.3.3) -, which through numerical experiments hasn’t been found to be a very restrictive condition; being more specific, we haven’t been able to find any combination of values for which the two access level curves cross more than once. Further, as is shown in the proofs in Appendix 1, if there exists at least one crossing point, then the number of crossing points must be odd. This implies that if qis not unique, then the curves from the two access level policies must cross at least three times. While imagining two concave curves that cross three times is not an impossible task, the shape required for such curves does not correspond with our observations. We believe there are two main situations where such complication could occur: First, when the average benefits for a given access level, and the shape of the effective demand distribution, are assumed to be independent; however, this assumption is essential to our model’s formulation since the average benefit is given by the proportion of patients that belong to each type, and therefore does not concern the problem studied here. Second, when the demand distribution has very “fat tails”, e.g., when the expected system’s utility function is relatively flat for a large range of values around the order quantity that maximizes the function. Such formulation could be appropriate if the size of the demand has high probabilities of being either very large or very small, and is left as an opportunity for further research. As a result of the above discussion, we will henceforth use the following assumption: A1: If qexists, it is unique. Proposition 1b, 1c, and 1d, provide the necessary and sufficient conditions for qto exist, 50 a1.2) Qς 1≤Qς 2<¯ Qς 1=q≤¯ Qς 2. a2) If ¯ Qς 1> q, then the possible orderings are: a2.1) Qς 1≤Qς 2≤q < ¯ Qς 1≤¯ Qς 2. a2.2) q < Qς 2≤Qς 1≤¯ Qς 1≤¯ Qς 2. a3) If ¯ Qς 1< q, then the possible orderings are: a3.1) Qς 1≤¯ Qς 1< q < Qς 2≤¯ Qς 2. a3.2) Qς 1≤Qς 2≤¯ Qς 2≤¯ Qς 1< q. a3.3) Qς 1<¯ Qς 1=Qς 2=¯ Qς 2< q. b) Suppose qdoes not exist. Then Qς 1≤Qς 2≤¯ Qς 2≤¯ Qς 1. There are a few interesting ideas on which it is worth elaborating based on Lemma 5; Figure 2.8 provides graphical examples to illustrate some of these different possible situations. Notice first that the crossing point between the two access level policies may occur: when the expected system’s utility is still increasing in both access level policies; when the expected system’s utility is already decreasing in both access level policies; or when the expected system’s utility is decreasing under the restricted access policy and increasing in the the full access policy. Another observation is that a necessary condition for Qς τ=¯ Qς τ>0 is that g > 0; such equality would imply that under the access level policy τ, there exists only one order quantity that achieves cost-effectiveness. Analyzing the Lemma by parts, for Lemma 5a1, as is formally shown in the proof, if ¯ Qς 1=q, then it is also true that the value of the curves is equal at an integer value. As result it is necessary that at least q=¯ Qς 2or q=Qς 2. Still, it is even possible that Qς 1=¯ Qς 1=q=Qς 2=¯ Qς 2; this situation requires two very specific circumstances to simultaneously occur. First, q=Qς 2=¯ Qς 2implies that under full access, the order quantity q maximizes the expected system’s utility achieving a value of zero. Similarly, Qς 1=¯ Qς 1=q can only mean that the expected system’s utility under the restricted access level policy 57 Figure 2.8: Ordering of the reference quantities (part 1) (a1.1) λ= 600; θ= 0.4; b1= 1; b2= 0.29; g= 0.5; δ= 0; c= 0.477 Qς 1= 118 <¯ Qς 1=q=Qς 2= 503 <¯ Qς 2= 722. (a1.2) λ= 600; θ= 0.95; b1= 1; b2= 0.12; g= 1.5; δ= 0; c= 0.9055 Qς 1= 538 < Qς 2= 589 <¯ Qς 1=q=¯ Qς 2= 629. (a2.1) λ= 600; θ= 0.4; b1= 1; b2= 0.3; g= 0.1; δ= 0; c= 0.4795 Qς 1= 39 < Qς 2= 300 < q = 442 <¯ Qς 1= 500 <¯ Qς 2= 725. (a2.2) λ= 600; θ= 0.95; b1= 1; b2= 0.96; g= 0.1; δ= 0.05; c= 0.65 Qς 1= 127 < Qς 2= 134 < q = 557 <¯ Qς 1= 902 <¯ Qς 2= 947. crosses the zero utility line from below at order quantity (q−ε), ε ∈(0,1) and then crosses it again from above at order quantity q, - notice that d(q−ε)e=q. This case of full equality is highly unlikely and implies that for both access level policies, the only order quantity that achieves cost-effectiveness is q; it requires that g > 0, that θapproaches 1, and that b2approaches b1. Further, taking into consideration that only a small subset of parameter combinations yield ¯ Qς 1=q, the relationship when the latter equality occurs is typically either: Qς 1<¯ Qς 1=q=Qς 2<¯ Qς 2, or Qς 1< Qς 2<¯ Qς 1=q=¯ Qς 2, as is shown in graphs (a1.1) and (a1.2) in Figure 2.8. 58 Figure 2.9: Ordering of the reference quantities (part 2) (a3.1) λ= 600; θ= 0.2; b1= 1; b2= 0.5; g= 0.5; δ= 0; c= 0.4795 Qς 1= 59 <¯ Qς 1= 250 < q = 382 < Qς 2= 484 <¯ Qς 2= 750. (a3.2-1) λ= 600; θ= 0.9; b1= 1; b2= 0.05; g= 0.5; δ= 0; c= 0.88 Qς 1= 436 < Qς 2= 578 <¯ Qς 2= 605 <¯ Qς 1= 613 < q = 625. (a3.2-2) λ= 600; θ= 0.9; b1= 1; b2= 0.06; g= 0.5; δ= 0; c= 0.88435 Qς 1= 439 < Qς 2=¯ Qς 2= 592 <¯ Qς 1= 610 < q = 622. (b) λ= 600; θ= 0.4; b1= 1; b2= 0.15; g= 0; δ= 0.2; c= 0.42 Qς 1=Qς 2= 0 <¯ Qς 2= 790 <¯ Qς 1= 872. The main difference between parts a2 and a3 is that for the former, Z(q, τ)>0, while for the latter Z(q, τ)<0, which may result in the aforementioned discontinuous range of order quantities. In Figure 2.8, graphs (a2.1) and (a2.1/a2.2) show situations when the joint effect of demand distribution and the average expected benefits for a given access level are relatively high relative to the transfer cost, c. It should be mentioned that the situation from Lemma 5-a2.2 couldn’t be exactly replicated numerically, but it is easy to show that if the cost-effectiveness threshold was positive and sufficiently large, instead of its currently assumed value of zero, then the shape of the graph (a2.1/a2.2) would fit into this case; 59 specifically if the cost-effectiveness threshold was raised to the horizontal red line, graph (a2.1/a2.2) corresponds to part a2.2 of Lemma 5. For part a3, graph (a3.1) shows the most common shape given that ¯ Qς 1< q. Graphs (a3.2-1) and (a3.2-2) both correspond to Lemma 5a3.2, the main distinction being that for the latter there is a single order quantity that satisfies cost-effectiveness under the full access policy. Part a3.3 requires a very specific combination of parameters, where there is a single order quantity that satisfies cost-effectiveness for full access, which coincides with the largest, but not unique, order quantity that satisfies cost-effectiveness under restricted access. Another important observation arising from Lemma 5 is that of dominance, i.e., the (weak) superiority of a particular access level policy in the expected social welfare utility - and consequently in the expected system’s utility - for any order quantity that satisfies the cost-effectiveness constraint. On one hand, restricted access weakly dominates full access under the situations that satisfy parts a3.2, a3.3, and b, of Lemma 5. In the first two of these cases, the intersection between the two access level policies occurs at an order quantity that is higher than that which maximizes the expected system’s utility under full access, and in such way that all order quantities above qyield a non cost-effective outcome. Through numerical experiments, we have observed that such situations are highly infrequent and occur when the fraction of type 1 patients is very high (above 0.9), the health benefit of type 2 patients is very low (below 0.1 b1), and the transfer cost cis high. For the situation from Lemma 5b, the result follows from Proposition 1 and the only condition is for the value of the outside option, or salvage value, to be higher than the value of health benefits for type 2 patients. On the other hand, full access weakly dominates restricted access under the situations that satisfy Lemma 5a2.2. This means that the crossing point between the two access level policies occurs at an order quantity that is lower than that which maximizes the expected system’s utility under restricted access. Again, through numerical experiments we have observed that this situation is not frequent either and requires the goodwill cost to be 60 positive and both the fraction of type 1 patients and the health benefits of type 2 patients to be very high, causing the two access level policies to almost overlap. For the rest of the combinations, there is no clear dominance and the optimal solution will depend on whether the inequalities in Lemma 5 are strong or weak, as well as on the value of the available budget. Summarizing, and perhaps most importantly, even though Lemma 5 may appear to create complexity by identifying a large variety of possible orderings, it is very useful in finding a structure to understand the drivers of the optimal decision making process. From this result, and assuming that there exists at least one feasible solution that yields Q∗ S,ς >0, we are now able to reduce the analysis of the optimal decisions to the relationship between the budget constraint, the minimum and maximum feasible quantities for full access, Qς 2and ¯ Qς 2, and the crossing point of the two access level policies, q. Theorem 1: Assume Qς τ, τ = 1,2 exist, and let Qς Γ≥Qς. When the integrated supply chain maximizes expected social welfare subject to budget and cost-effectiveness constraints, the optimal solution is as follows. a) Suppose qexists. a1) min{¯ Qς 2, Qς Γ} ≤ max{q , Qς 2}, is a necessary condition for τ∗ S,ς = 1 and Q∗ S,ς = min{Qς Γ,¯ Qς 1}to be an optimal solution. a2) min{¯ Qς 2, Qς Γ}<max{q , Qς 2}, is a necessary and sufficient condition for τ∗ S,ς = 1 and Q∗ S,ς = min{Qς Γ,¯ Qς 1}to be the unique optimal solution. a3) min{¯ Qς 2, Qς Γ} ≥ max{q , Qς 2}is a necessary condition for τ∗ S,ς = 2 and Q∗ S,ς = min{Qς Γ,¯ Qς 2}to be an optimal solution. a4) Jointly satisfying (min{¯ Qς 2, Qς Γ}> q) and (min{¯ Qς 2, Qς Γ} ≥ Qς 2), is a necessary and sufficient condition for τ∗ S,ς = 2 and Q∗ S,ς = min{Qς Γ,¯ Qς 2}to be the unique optimal solution. a5) min{¯ Qς 2, Qς Γ}=q≥Qς 2is a necessary and sufficient condition for the decisionmaker to be indifferent between (τ∗ S,ς = 1 and Q∗ S,ς = min{Qς Γ,¯ Qς 1}) versus (τ∗ S,ς = 2 and 61 Q∗ S,ς = min{Qς Γ,¯ Qς 2}). b) Suppose qdoes not exist; then τ∗ S,ς = 1 and Q∗ S,ς = min{Qς Γ,¯ Qς 1}. Theorem 1 shows the conditions under which the optimal access level will be either restricted (τ∗ S,ς = 1), or fully inclusive (τ∗ S,ς = 2), and the corresponding optimal order quantity of drugs. Note first that Theorem 1-a1 and -a3 are the result of Theorem1-a2, -a4, and -a5. It is also easier to see the relevance of the (weak) dominance relationships we discussed after Lemma 5. Namely, Lemma 5-a1.2, -a3.2, -a3.3 correspond to Theorem 1-a1, and as a result also correspond to either Theorem 1-a2 or -a5. Similarly, Lemma 5-a2.2 corresponds to Theorem 1-a3, and as a result to either Theorem 1-a4, or -a5. Finally, Lemma 5-b corresponds to Theorem 1-b. Since the latter situations are less interesting from a joint decision making perspective, none of the cases where strong dominance exists is represented in Figures 2.10 and 2.11, but the interested reader can verify in the corresponding graphs shown in Figures 2.8 and 2.9 that regardless of the magnitude of the available budget, the access level will remain constant. Figures 2.10 and 2.11 is then used to assist the process of understanding how the solution changes in many of the remaining cases once the budget constraint is included. One of the interesting observations is that when the budget is sufficiently high and the value of the outside option is lower than the expected health benefits for type 2 patients, full access level may be optimal for the social welfare maximizer even if the average health benefit received by patients of type 2 is lower than the cost of the drug, e.g., Figure 2.10, graph (2); the explanation is that the expected health benefits achieved by patients of type 1 subsidize those belonging to the second type, making aggregate cost-effectiveness possible. In terms of indifference, graphs (1) and (12) show examples where the decision maker is indifferent between both choices of access level, but the drivers and implications are very different. On one hand, in graph (1) the budget constraint is the key driver of the decision 62 Figure 2.10: Optimal decision making under expected social welfare maximization (part 1) lrepresents Qς Γ (1) λ= 600; θ= 0.4; b1= 1; b2= 0.29; g= 0.5; δ= 0; c= 0.477; τ∗ S,ς =1 or 2, and Q∗ S,ς =¯ Qς 1=Qς Γ (2) λ= 600; θ= 0.4; b1= 1; b2= 0.29; g= 0.5; δ= 0; c= 0.477; τ∗ S,ς = 2 and Q∗ S,ς =¯ Qς 2 (3) λ= 600; θ= 0.45; b1= 1; b2= 0.65; g= 0.5; δ= 0; c= 0.3; τ∗ S,ς = 2 and Q∗ S,ς =Qς Γ (4) λ= 600; θ= 0.45; b1= 1; b2= 0.65; g= 0.5; δ= 0; c= 0.3; τ∗ S,ς = 2 and Q∗ S,ς =Qς Γ (5) λ= 600; θ= 0.7; b1= 1; b2= 0.65; g= 0; δ= 0.2; c= 0.85; τ∗ S,ς = 1 and Q∗ S,ς =Qς Γ (6) λ= 600; θ= 0.7; b1= 1; b2= 0.65; g= 0; δ= 0.2; c= 0.85; τ∗ S,ς = 2 and Q∗ S,ς =Qς Γ 63 Figure 2.11: Optimal decision making under expected social welfare maximization (part 2) lrepresents Qς Γ (7) λ= 600; θ= 0.3; b1= 1; b2= 0.9; g= 1; δ= 0; c= 0.65; τ∗ S,ς = 1 and Q∗ S,ς =¯ Qς 1 (8) λ= 600; θ= 0.3; b1= 1; b2= 0.9; g= 1; δ= 0; c= 0.65; τ∗ S,ς = 2 and Q∗ S,ς =Qς Γ (9) λ= 600; θ= 0.3; b1= 1; b2= 0.4; g= 1; δ= 0; c= 0.5; τ∗ S,ς = 2 and Q∗ S,ς =Qς Γ (10) λ= 600; θ= 0.3; b1= 1; b2= 0.4; g= 1; δ= 0; c= 0.5; τ∗ S,ς = 2 and Q∗ S,ς =¯ Qς 2 (11) λ= 600; θ= 0.95; b1= 1; b2= 0.12; g= 1.5; δ= 0; c= 0.9055; τ∗ S,ς = 1 and Q∗ S,ς =Qς Γ (12) λ= 600; θ= 0.95; b1= 1; b2= 0.12; g= 1.5; δ= 0; c= 0.9055; τ∗ S,ς = 1 and Q∗ S,ς =¯ Qς 1=¯ Qς 2 64 maker’s indifference since moving it slightly in any direction would yield a unique solution. This situation tends to occur when the fraction of type 1 patients is medium to low; in such cases, the decision maker will need to choose between providing high service level to a limited fraction of the population, versus providing low service level to the whole patient population. On the other hand, in graph (12) the key driver of the decision maker’s indifference is the cost-effectiveness constraint. This situation tends to occur when the fraction of type 1 patients is very high and the transaction cost, c, is also high; in such cases, there will be no major difference between restricting or not access to type 2 patients. Finally, note that by moving the budget constraint appropriately in some of the other graphs, such that Qς γ=q, the indifference issue would also arise; this brings the attention to the fact that under expected social welfare maximization, the decision maker may only be indifferent when Z(q, τ)≥0, τ = 1,2, i.e., for graphs (7)-(10) the optimal solution is unique regardless of the available budget because the crossing point of the curves occurs at a point which is not cost-effective. Finally, when only one access level satisfies the cost-effectiveness constraint, the solution is straightforward and given in Corollary 1. Corollary 1: When the integrated supply chain maximizes social welfare subject to budget and cost-effectiveness constraints: a) Assume Qς 1≤Qς Γexists and Qς 2does not exist; then τ∗ S,ς = 1 and Q∗ S,ς = min{Qς Γ,¯ Qς 1}. b) Assume Qς 2≤Qς Γexists and Qς 1does not exist; then τ∗ S,ς = 2 and Q∗ S,ς = min{Qς Γ,¯ Qς 2}. Case 2ς: Maximizing the system’s expected utility function In this section we solve the problem of maximizing expected system’s utility function as might be the case for a private insurance company. The problem faced by the single decision 65 maker under this setting is: max (Q,τ)Z(Q, τ) = (B(τ)−δ+g)A(Q, τ)−(c−δ)Q−gλF(τ) subject to: cQ ≤Γ −cQ +B(τ)A(Q, τ) + δ(Q−A(Q, τ)) −g(λF(τ)−A(Q, τ)) ≥0(2.3.8) Define Q∗ H,ς and τ∗ H,ς to be the optimal order quantity and prescription policy threshold, respectively, when maximizing the system’s expected utility function. Let Q ς H={Qς Γ, Qς 1, Qς 2}, denote the set of possible optimal order quantities under this setting, recalling that Qς Γ= Γ c,stands for the order quantity when the budget constraint is binding, while Qς τ∈arg max Q {Z(Q, τ)}, τ = 1,2, is the order quantity that maximizes the system’s utility function contingent on τ. From Lemma 2, the system’s expected utility function is concave for a given τ, and we can use the method of finite differences to obtain: Qς τ= max Q P(Q;λF(τ)) ≥c−δ B(τ)−δ+g, τ = 1,2.(2.3.9) Before deriving the optimal access and service levels for this case, some intermediate results are needed. Proposition 2: Define ˜c={c|Z(Qς 1,1) = Z(Qς 2,2)}. a) ˜cexists ⇐⇒ qexists. 66 solution. a3) Jointly satisfying Qς Γ≥ bQς 2cand c < ˜cis a sufficient condition for τ∗ H,ς = 2 and Q∗ H,ς = bQς 2cto be the unique optimal solution. a4) Jointly satisfying (Qς Γ≥ bQς 2c) and (c= ˜c), is a necessary and sufficient condition for the decision-maker to be indifferent between: [τ∗ H,ς = 1 and Q∗ H,ς =bQς 1c] and [τ∗ H,ς = 2 and Q∗ H,ς =bQς 2c]. a5) If bQς 2c> Qς Γ≥max{q , Qς 2}and c < ˜c, then the result is ambiguous. b) Suppose qdoes not exist; then τ∗ H,ς = 1, and Q∗ H,ς = min{Qς Γ,bQς 1c}. Corollary 2: When the integrated supply chain maximizes the system’s expected utility subject to budget and cost-effectiveness constraints: a) τ∗ S,ς = 1 is a sufficient, but not necessary, condition for τ∗ H,ς = 1. b) τ∗ S,ς = 2 is a necessary, but not sufficient, condition for τ∗ H,ς = 2. c) Q∗ H,ς ≤Q∗ S,ς. One of the key implications from the above results is that the combination of parameters for which the full access policy is chosen decreases with respect to the social welfare maximization case. The result is obtained by comparing Theorem 2a2 versus Theorem 1a3. For example, under social welfare maximization, as the budget increased, the probability of choosing the full access policy was only limited by cost-effectiveness; however, under system’s expected utility maximization, even as Γ → ∞, full access policy necessarily requires c≤˜c, including the indifference case. Figures 2.16 and 2.17 can facilitate the processing of the analytical results by observing how and when did the optimal decision change in relation to Figures 2.10 and 2.11. Graphs (2), (3), (6), (9), (10), and (12) all show strict decreases in the access level policy, while for graph (1), choosing restricted access has become the unique optimal solution; it’s worth mentioning that from all the latter, only in graph (3) the budget acts as a limiting constraint. Going deeper, graphs (3) and (8) represent the situations 73 Figure 2.16: Optimal decision making under system’s expected utility maximization (part 1) lrepresents Qς Γ (1) λ= 600; θ= 0.4; b1= 1; b2= 0.29; g= 0.5; δ= 0; c= 0.477; τ∗ H,ς =1 and Q∗ H,ς =Qς 1 (2) λ= 600; θ= 0.4; b1= 1; b2= 0.29; g= 0.5; δ= 0; c= 0.477; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 (3) λ= 600; θ= 0.45; b1= 1; b2= 0.65; g= 0.5; δ= 0; c= 0.3; τ∗ H,ς = 1 and Q∗ H,ς =Qς Γ (4) λ= 600; θ= 0.45; b1= 1; b2= 0.65; g= 0.5; δ= 0; c= 0.3; τ∗ H,ς = 2 and Q∗ H,ς =Qς 2 (5) λ= 600; θ= 0.7; b1= 1; b2= 0.65; g= 0; δ= 0.2; c= 0.85; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 (6) λ= 600; θ= 0.7; b1= 1; b2= 0.65; g= 0; δ= 0.2; c= 0.85; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 74 Figure 2.17: Optimal decision making under system’s expected utility maximization (part 2) lrepresents Qς Γ (7) λ= 600; θ= 0.3; b1= 1; b2= 0.9; g= 1; δ= 0; c= 0.65; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 (8) λ= 600; θ= 0.3; b1= 1; b2= 0.9; g= 1; δ= 0; c= 0.65; τ∗ H,ς = 1 and Q∗ H,ς =Qς Γ (9) λ= 600; θ= 0.3; b1= 1; b2= 0.4; g= 1; δ= 0; c= 0.5; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 (10) λ= 600; θ= 0.3; b1= 1; b2= 0.4; g= 1; δ= 0; c= 0.5; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 (11) λ= 600; θ= 0.95; b1= 1; b2= 0.12; g= 1.5; δ= 0; c= 0.9055; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 (12) λ= 600; θ= 0.95; b1= 1; b2= 0.12; g= 1.5; δ= 0; c= 0.9055; τ∗ H,ς = 1 and Q∗ H,ς =Qς 1 75 defined by Theorem 2a5. Intuitively, for Theorem 2a5 as cincreases, Qς Γdecreases, and c approaches ˜c, increasing the the likelihood that access will be restricted; having a large gap between b2and chas higher likelihood of inducing full access. Taking a graphical approach based on Figures 2.16 and 2.17, it can be observed that while in (3) the budget is not large enough to justify full access, it is indeed so for graph (8) despite being forced to order less than the quantity that maximizes the unconstrained problem; the latter graph is also an appropriate example for why Theorem 2a3 does not provide a necessary condition. The other change of interest is that of the order quantity. It is easy to see from Figures 2.16 and 2.17 how the optimal order quantity is (weakly) reduced in relation to the social welfare maximization case. The most frequent situation when Q∗ H,ς =Q∗ S,ς given τ∗ H,ς = 2, is while being in the region defined by Theorem 2a5 such that 0 ≤Z(Qς 1,1) ≤Z(Qς Γ,2) ≤ Z(Qς 2,2); similarly, the most frequent situation when Q∗ H,ς =Q∗ S,ς given τ∗ H,ς = 1, is when 0≤Z(Qς Γ,2) ≤Z(Qς 1,1). 2.4 Extension for more than two types of patients In this section, we describe a basic heuristic for solving the problem when there are I > 2 types of patients. The previous definitions are directly extended to this situation, i.e., let birepresent the health benefits for type ipatients, and assume b1< b2< . . . bI. It is true that for most situations within the context described, the number of patient categories for which a drug can provide relevant health benefits is not expected to be very large. As a result, doing a sequential pairwise comparison wouldn’t be a time consuming task. This would imply comparing types 1 and 2 and finding an optimal solution; then comparing the optimal access level between them against including type 3 patients, and so on. However, for the case when cis fixed, we are able to define more efficient algorithms contingent on the problem’s structure. We will do so first for the social welfare maximizer, and then for the total utility maximizer. 76 2.4.1 Maximizing the expected social welfare given I > 2types of patients Based on the analysis from §2.3, we will use the additional definition for the crossing point between the expected social welfare curves of two access level policies. Definition 1: Let qi,j,1≤i<j≤I, be a positive order quantity such that S(qi,j, i)≥ S(qi,j, j); and S(qi,j + 1, j)> S(qi,j + 1, i). When the decision maker is trying to maximize the expected social welfare, Lemma 6 provides a key result which simplifies the calculations. Lemma 6: For 1 ≤i≤(I−2), min{qi,i+1, qi+1,i+2} ≤ qi,i+2 ≤max{qi,i+1, qi+1,i+2}. The usefulness of Lemma 6 is that not all the crossing points need to be calculated in order to determine the access level with the highest social welfare for any given order quantity. To be more specific, only the crossing points between consecutive access levels are necessary. Also, recall from Proposition 1 and Theorem 1, that when there is no crossing point between the curves of two policies, then the policy with greater access is dominated by the one with more restricted access. Since such result is now trivial, this section is only concerned with the situations where further analysis is necessary to determine the optimal solution. Next, Lemma 7 gives a condition which may further reduce the set of potential optimal solutions for the access level. Lemma 7: Let 1 < i < I. a) if qi,i+1 < qi−1,i+1 < qi−1,i, then the expected social welfare under access level iis dominated by either access levels (i−1) or access level (i+ 1), for any Q. b) otherwise, if qi−1,i ≤qi−1,i+1 ≤qi,i+1, then no access level (i−1), i, and (i+ 1), is 77 Figure 2.18: Three Types of patients (1) and (2) λ= 600; f(1) = f(2) = f(3) = 1/3; b1= 1; b2= 0.4; b3= 0.3; g= 0.5; δ= 0; c= 0.3 (3) and (4) λ= 600; f(1) = f(2) = f(3) = 1/3; b1= 1; b2= 0.4; b3= 0.3; g= 0.5; δ= 0; c= 0.6 dominated by the other two. When Lemma 7a is satisfied for any ibetween 1 and I, then access level ican be discarded from the calculations. However, it must be noted that we were neither able to generate a combination of parameters that corresponds with this case, nor prove that this case can’t occur; for this reason, it is included in the results, but will not be the focus of the analysis henceforth. In contrast, Lemma 7b represents the typical behavior of the expected social welfare and system’s utility curves under different access level policies. Figure 2.18 provides a graphical representation for two different costs, c. Under this situation, before considering the cost-effectiveness and budget constraints, every access level policy remains a feasible candidate for optimality under expected social welfare maximization. As a result, we propose an algorithm to reach the optimal solution without having to solve completely for every access level policy. The main value of the algorithm is that it allows the results of §2.3 to 78 be extended to multiple types. While a trial and error approach would also be feasible, at this point it should be clear that the different ways in which the social welfare and total expected utility curves shift relative to changes in the parameters and decision variables makes it highly complicated to correctly anticipate the system’s dynamics, and therefore its optimal solution, without full enumeration. Proposition 4: Suppose qi−1,i ≤qi,i+1, for 1 < i < I. The solution algorithm for maximizing expected social welfare is as follows. 1) Set qk−1,k = ˆq, where ˆqis given by equation (2.4.1). 2.1) If qk−1,k = ˆq, calculate Qς k, and ¯ Qς k, and go to step 3.1. 2.2) Otherwise, calculate qk−1,k,Qς k, and ¯ Qς k, and go to step 3.1. 3.1) If min{¯ Qς k, Qς Γ} ≥ max{qk−1,k , Qς k}, then τ∗ S,ς =kand Q∗ S,ς = min{Qς Γ,¯ Qς k}is an optimal solution. Go to step 4.1. 3.2) Otherwise, set k= (k−1), and go back to step 2. 4.1) If (min{¯ Qς k, Qς Γ}> qk−1,k) and (min{¯ Qς k, Qς Γ} ≥ Qς k), then τ∗ S,ς =kand Q∗ S,ς = min{Qς Γ,¯ Qς k}is the unique optimal solution. END. 4.2) Otherwise, the decision maker is indifferent between (τ∗ S,ς = (k−1) and Q∗ S,ς = min{Qς Γ,¯ Qς k−1}) versus (τ∗ S,ς =kand Q∗ S,ς = min{Qς Γ,¯ Qς k}). END. ˆq,       max{q∈ {q1,2, . . . , qI−1,I} | q≤Qς Γ},if q1,2≤Qς Γ q1,2otherwise (2.4.1) Proposition 4 integrates the results from Lemma 6 and Lemma 7 with the constraints faced by the single decision maker. Figure 2.19, is useful in observing how the algorithm works. Note that graphs (1), (3) and (5) only vary in the (decreasing) value of the budget constraint; the same applies to graphs (2), (4), and (6). For example, the algorithm path 79 Figure 2.19: Optimal decision making under expected social welfare maximization for I= 3 lrepresents Qς Γ (1), (3), and (5): λ= 600; f(1) = f(2) = f(3) = 1/3; b1= 1; b2= 0.4; b3= 0.3; g= 0.5; δ= 0; c= 0.3 (2), (4), and (6): λ= 600; f(1) = f(2) = f(3) = 1/3; b1= 1; b2= 0.4; b3= 0.3; g= 0.5; δ= 0; c= 0.6 (1) ˆq=q2,3;τ∗ S,ς = 3 and Q∗ S,ς =Qς Γ (2) ˆq=q2,3;τ∗ S,ς = 2 and Q∗ S,ς =¯ Qς 2 (3) ˆq=q2,3;τ∗ S,ς = 2 or 3, and Q∗ S,ς =Qς Γ (4) ˆq=q2,3;τ∗ S,ς = 2 and Q∗ S,ς =Qς Γ (5) ˆq=q1,2;τ∗ S,ς = 1 or 2, and Q∗ S,ς =Qς Γ (6) ˆq=q1,2;τ∗ S,ς = 1 or 2, and Q∗ S,ς =Qς Γ followed in graph (1) is steps: 1 →2→3.1→4.1, which results in a unique solution. Instead, graph (2) follows the path: 1 →2.1→3.2→2.2→3.1→4.1, which also results in a 80 unique solution; the difference is that the algorithm had to loop because the cost-effectiveness constraint was not satisfied for the feasible range of order quantities. The path followed in graph (4) is the same. Graph (3) shows the situation where the decision maker is indifferent between two access levels, following the path: 1 →2.1→3.1→4.2. Finally, graphs (5) and (6) show the case where q1,2> Qς Γ, and therefore ˆq=q1,2; the algorithm then follows the path 1 →2.1→3.1→4.1 which results in a unique solution. 2.4.2 Maximizing the system’s expected utility function given I > 2types of patients Now we look at the problem when the decision maker is concerned with maximizing the system’s expected utility. Just like in §2.4.1 we had used the crossing point between the utility curves to simplify the analysis, in this part we use the key threshold that had been derived earlier in §2.3.3 to determine which access level policy would achieve a higher expected utility for the system. Definition 2: Let ˜ci,j =cZ(Qς i, i) = Z(Qς j, j); 1 ≤i<j≤I. Lemma 8: For 1 < i < I, ˜ci−1,i >˜ci,i+1. Lemma 8 can be inferred from the previously stated observation that ˜ci−1,i →bifor most parameter combinations; since biis decreasing in i, then the required selling price for equating the expected utility under two consecutive access level policies is also decreasing in i. The main implication is that if c > ˜ci,i+1, then c > ˜cj,j+1, 1 ≤i < j < I,i.e., when the selling price is sufficiently high for the more restricted policy to dominate in a consecutive pairwise combination, then that restricted policy also dominates the rest of the more inclusive access level policies. The result can be observed graphically in Figures 2.20 and 2.21, and is further 81 Figure 2.20: Changes in cfor I= 3 (part 1) λ= 600; f(1) = 0.25; f(2) = 0.25; f(3) = 0.5; b1= 1; b2= 0.6; b3= 0.2; g= 0; δ= 0 (1)c= 0.8; (2)˜c1,2= 0.60241; (3)c= 0.4; (4)˜c1,3= 0.33342; (5)˜c2,3= 0.19895; (6)c= 0.05. expanded in Lemma 9. Figure 2.21: Changes in cfor I= 3 (part 2) λ= 600; f(1) = 1/9; f(2) = 2/9; f(3) = 2/3; b1= 1; b2= 0.4; b3= 0.15; g= 1; δ= 0 (1)c= 0.45; (2)˜c1,2= 0.38090; (3)c= 0.25; (4)˜c1,3= 0.20365; (5)˜c2,3= 0.14416; (6)c= 0.1. 82 mined; such assumption best reflects those situations where external reference pricing is used. From a methodological perspective, the first contribution is the finding of a unique crossing point between the expected social welfare for the different levels of access that allows us to quickly determine the optimal order quantity and access level under expected social welfare maximization. Comparative statics and an extensive discussion has been included to explain the direction of such threshold as a function of the cost and benefit parameters. Second, for the case when the decision maker maximizes his expected net utility we achieve a similar result by finding a threshold transfer price such that any transaction price higher than the threshold will result in restricted access; through numerical experiments, we have observed this value to be very close to the marginal health benefit of the type of patients with lower health benefits. An additional, interesting property is that each of these thresholds exists if and only if the other threshold exists as well, despite the fact that social welfare does not depend on cost. Third, based on these two thresholds, we provide an efficient solution process which relates to the price and quantity newsvendor model studied in operations management; our contribution to earlier analyses is the determination of the optimal solution when the choice of the optimal access level (which serves the same purpose as the retail price in the operations literature) is discrete, the decision space is constrained by absolute and relative cost constraints, and when the objective function can be to either maximize the expected net utility (as traditional models in operations management do) or maximize social welfare (which is the mandate for many of the relevant players in our context). Finally, we have provided a heuristic for finding the optimal solution when there are more than two patient types under minor assumptions which represent most of the feasible space. From a policy-making perspective, we first identify situations of strong dominance of a given access level policy, independent of the available budget. An important observation is that a social welfare maximizer is prone to subsidizing patients whose expected benefits 89 are lower than the transfer price as long as the available budget is sufficiently high, while maximizing the health payer’s expected utility will not do so. This implies that markets where external reference pricing is used to determine prices will be highly dependent on their available budget (under social welfare maximization) and on relative health benefits, demand size, and demand uncertainty (under Health’s utility maximization) to determine the optimal policy, since these were the main drivers of the results of the two cases, correspondingly, in §2.3.3. Also, we find that when access level is restricted under social welfare maximization, then it will be restricted as well under net utility maximization; however, the opposite is not necessarily true. Such situations occur due to either high transfer costs (higher than the threshold value) relative to the benefit of the lower type patients, or to relatively low budget constraints. Moreover, we find that the optimal order quantity is weakly reduced under expected net utility maximization, even in the situations when the budget is infinitely high or when the access decision remains unchanged relative to social welfare maximization. An interesting and necessary extension relates to the response function from the manufacturer’s perspective. Endogenously setting the price, or entering into some risk sharing agreement are the focus of Chapter 3. Additionally, the extension of our results to more general demand distributions would be a useful validation of the intuition here provided; however, we do expect the results to be qualitatively consistent for IGFR distributions. 90 Chapter 3 Analyzing the value of three endogenous contracting mechanisms in the joint access and coverage problem in health care 3.1 Introduction In Chapter 2 we have analyzed the decision-making process of a health-payer under a fixed transfer price. However, pharmaceutical manufacturers also play an important role is setting the conditions under which transactions will occur between themselves and the healthpayers. Motivated either by the increasing pressures held by the payers, or by selfish profit maximization, a variety of mechanisms have been attempted in order to modify the status quo. The financial risks associated to demand uncertainty and the asymmetry between the health benefits claimed by the pharmaceutical manufacturers and those acknowledged by the health-payers are two key reasons for the introduction of new drugs aimed at treating chronic conditions to be rejected, delayed, or accepted under terms that negatively impact the player with the lowest bargaining power. On top of the budget constraints that health91 payers may have, the health benefit value they use in their calculations of cost-effectiveness, expected social welfare, and expected total utility, has a high relevance on both the access and service levels under which the drug is commercialized (if at all). By delaying introduction, health-payers wish to either reduce their uncertainty about the drug’s performance in clinical practice, or negotiate a lower selling price with the manufacturer. Said manufacturer, at the initial stage of negotiation has the option of either decreasing the selling price or accepting to commercialize a lower sales volume of the drug. If neither alternative is accepted by the manufacturer, then she will collect more evidence hoping to increase her bargaining power against health-payers in the future. Some of the main problems, however, are that until the drug is accepted for introduction, the manufacturer is losing revenues, the patent clock is ticking, and the patients are not able to receive what is supposed to be the most appropriate treatment. As a result risk sharing contracts have received increasing attention. Particularly in the United Kingdom, the discussion in academic and political environments has been very active. Pouvourville (2006) discussed the attractiveness of risk sharing contracts in managing the uncertainty surrounding a product’s performance in real life and the credibility of the claims by the manufacturers, while also providing some predictability for such manufacturers. Carapinha (2008) highlighted the importance of integrating clinical, quality of life, and financial outcome measures into a risk-sharing agreement as well as the challenges of patient compliance and inefficient delivery of health care services. He also comments on the inconclusive evidence on the impact of risk sharing agreements on an individual patient’s clinical and quality of life outcomes, and on their effectiveness in containing pharmaceutical expenditure. This chapter attempts to add additional insight into the value and limitations of three contracting mechanisms proposed by the manufacturer. First we analyze the impact of endogenizing the transaction cost of the drug, and find 92 situations when a health-payer maximizing his expected utility may be able to negotiate a lower price, and achieve higher social welfare, than a health-payer who maximizes expected social welfare. Second, we characterize the conditions under which Pharma is willing to build capacity above Health’s initial commitment, and show that this type of contracts result in a weak increase in both access level and expected social welfare. Third, we propose a new performance-based mechanism that partially reduces the negative effects of asymmetric beliefs between Pharma and Health. The rest of the chapter proceeds as follows. In §3.2, additional literature specific to the proposed contracts is briefly addressed to complement the one presented in the previous chapter. Section 3.3 solves the endogenous price-only contracts. Section 3.4 relaxes the single ordering assumption and solves the capacity buffer contract under an exogenously set transfer price. In section 3.5 Pharma offers a performance-based contract to Health to deal with belief asymmetry. Concluding remarks are offered in §3.6. 3.2 Literature Review The search for alternatives to manage demand and health outcome uncertainty in order to make a better use of the available and continuously decreasing resources, has produced a large volume of work in recent years, both theoretical and applied. From an applied perspective, Pugatch, Healy, and Chu (2010) provide a survey of 27 agreements between manufacturers and health payers implemented over the last two decades across five countries (United Kingdom, Italy, Australia, Germany, and the United States) for drug treatments that would have otherwise been rejected. They identify 4 mechanisms: cost caps and rebates (which are driven by price), and patient monitoring and patient compliance (which are driven by performance). It is worth mentioning that 16 out of the 27 agreements included some form of rebate in the contract’s conditions. Espin, Rovira and Garcia (2011) analyze risk sharing schemes in 93 Europe for oncology products, which they categorize as financially-based schemes, i..e, pricevolume agreements with paybacks or price reductions, and outcome-based schemes. They find some form of risk sharing scheme in 7 countries: Portugal, France, United Kingdom, Italy, Slovenia, Germany, and Lithuania. The main objectives of the agreements observed were to control the budget, get additional data, or finance cost-effective medicines. The contract proposed in §3.3 aims to represent the situations where reference pricing is not used to determine the transfer price between the manufacturer and the health-payer. We base our model on the analysis of price-only contracts by Lariviere and Porteus (2001) where the conditions for the manufacturer’s objective function to be unimodal in the selling price are defined in Theorem 1 (p. 296). The key differences are that in their model, they consider the retail price to be fixed, and there are no absolute nor relative budget constraints for the downstream party (the retailer in their model, the health-payer in ours). As a result, our task is to develop a method for efficiently finding the optimal transfer price, incorporating the aforementioned factors. By doing this, we also expand the works of Salinger and Ampudia (2011) and Kocabiyikoglu and Popescu (2011), providing a supply chain perspective to the analysis of the price and quantity newsvendor model, where the the upstream party is able to determine the transfer price. The main contribution is the added visibility of the relationship between the transfer price-setting process and the downstream party’s optimal decisions, as a function of the objective function and constraints faced by the latter. The contract proposed in §3.4 goes back to the exogenous price assumption, but relaxes the single order opportunity constraint of the classic newsvendor setting by allowing the upstream party to overproduce, and therefore letting the downstream party to order above and beyond its initial order quantity. The contract has its roots in two well known contracts in the supply chain coordination literature. First, the quantity flexibility contracts (Tsay, 1999), where the buyer sends a purchasing signal of size qwell before observing demand, 94 and the manufacturer builds a stock of q(1 + α), the buyer is committed to purchase at least q(1 −w), and the contract parameters are the selling price and the sales range parameters α > −w, and w∈[0,1]. Second, the buyback contract (Pasternack, 1985) where the manufacturer chooses selling price wand buyback rate b, which is the price paid by the manufacturer to the buyer for every unit of overstock at the end of the demand period. Both contracts coordinate the supply chain in a wide array of scenarios, but our contract is different both in its main objective and its decision variables. The proposed contract does not explicitly guarantee a minimum capacity on the part of the manufacturer because symmetric information is assumed regarding the manufacturer’s production costs; therefore the buyer is able to anticipate the manufacturer’s optimal capacity. Additionally, in our model the capacity is not necessarily bounded by the manufacturer’s incentive compatibility constraint as in the quantity flexibility, but rather may be limited by Health’s constraints. Finally, while selling KTunits to the buyer at price wand repurchasing the excess units at price b is similar to our approach of the buyer incurring a penalty for every unit ordered above its initial order quantity, it is worth noting that the only decision variable that the manufacturer has in our model is the total capacity, as price is considered to be exogenously determined. As a result, our contract is not directly aimed at coordinating the inventory decision, but rather seeks to understand the conditions under which the manufacturer will voluntarily build inventory above and beyond the health payer’s initial order quantity when external reference pricing is used as a mechanism to determine transfer payments between the players. Regarding the contract proposed in §3.5, which is a performance-based contract, Guajardo, Cohen, Kim and Netessine (2012) study the ability of performance based contracts in a general setting to increase product reliability, and find that such reliability is increased due to more frequent and more diligent maintenance activities induced by the optimal contract. Our model, rather than allowing the exertion of efforts that may affect the distribution of the realized health benefits, assumes that the manufacturer has private knowledge about the 95 expected performance of the drug, and therefore we focus not on the ability to modify the value of the product, but rather on the manufacturer’s ability to signal the ability to the buyer. From the works of supply chain coordination using the newsvendor model, under price-dependent demand in a seller-buyer relationship, Emmons and Gilbert (1998) show that buy-back contracts with a fixed buy-back rate do not coordinate the chain; further, considering a fixed payment per unit sold, buy-back contracts coordinate the chain but allocate zero profit to the supplier (Marvel and Peck, 1995; Bernstein and Federgruen, 2005). Bernstein and Federgruen (2005) show that a buy-back contract coordinates the chain under arbitrary profit allocation only if the buy-back rate and the wholesale price are adjusted as a function of the retail price in what is referred to as “contingent buy-backs” or “discount pricing”. Cachon and Lariviere (2005) show that revenue sharing may coordinate the chain when the buyer selects the retail price, but similar to buy-backs, an arbitrary allocation of profit requires the contract parameters to be contingent on the selling price. The main limitation of these models within our setting is that they assume the “selling price” to be a deterministic parameter while in our context after the prescription policy threshold is set, the resulting health benefits are a random variable. This would be equivalent to being able to select only the expected selling price, and setting the contract parameters accordingly. This distinction about a random selling price - health benefits in our model - implies that the player whose payoff function depends on the realized price holds a higher risk in the contract, and that the contract parameters would need to be modified after the true price is revealed. Needless to say, this raises concerns on how such a mechanism could be successfully implemented in our setting. By considering different objective functions for the downstream party - the health-payer - under a set of constraints relevant to the decision of introducing a new drug, we provide additional light into the theoretical reach and applicability of risk sharing contracts in health care. This type of contracts has received increasing attention in recent years by the health 96 economics community. Barros (2011) looks at the relationship between a pharmaceutical manufacturer and a health-provider’s prescription behavior assuming a binary health outcome and patient heterogeneity; he studies a risk sharing contract where the manufacturer is reimbursed for the drug only when the treatment is successful, which leads to high list prices and a higher than efficient prescription behavior, even though the latter effect may be alleviated by appropriately setting a revision cost. The main differences with our model is that Barros (2011) does not incorporate demand uncertainty and the prescriber experiences no risk in the contract. On a related paper, Zaric and Xie (2009) analyze two contracts in a two-period setting where the manufacturer sets the price for a drug seeking formulary listing and exerts promotional effort that deterministically shapes the demand curve: in one contract, the drug is listed in the payer’s formulary during period 1 and delisted in the next period if cost-effectiveness is not achieved, and in the second contract, which is the most relevant to our work, the manufacturer pays a rebate to the health-payer in each period that cost-effectiveness is not achieved where the rebate amount is such that the payer’s cost-effectiveness constraint binds. They find that no contract dominates, and provide a numerical analysis to observe the effects of uncertainty, the willingness to pay threshold, and the associated costs (or savings) derived from the drug’s introduction. The first distinction in our model is that we allow the size of demand to be uncertain, which creates an inventory risk for the health-payer that becomes relevant for both his objective function and his cost-effectiveness constraint. Second, our model does not consider the manufacturer’s ability to influence the size of total incoming demand. Third, we explicitly model information asymmetry with respect to the expected health outcome and allow the manufacturer to take advantage of its informational advantage through the parameters of the contract presented in §3.5. Fourth, the performance-based contract proposed here is similar in that a rebate is also offered, but in our case the rebate is a fixed amount given by a per unit rebate rate set by the manufacturer multiplied by the order quantity of the health-payer, while in Zaric and Xie (2009), the rebate is such that realized net monetary benefits are zero (i.e., 97 if realized outcomes are below the minimum level of acceptance, the rebate is such that the cost-effectiveness constraint binds). 3.3 Endogenous price-only contracts In this section we model the relationship between Pharma and Health when the former is able to endogenously select the transfer price, which is assumed to be the only contract parameter. This kind of contracts reflects more appropriately those cases where a negotiation process occurs between the manufacturer and the health-payer so that the former observes the parameters used by the latter in his calculations. We denote this setting with the symbol ηfor endogenous. We will continue to use the notation presented in Chapter 2, and introduce additional notation as needed. Pharma sells each unit of the drug to Health at endogenously selected price w, in order to maximize its utility function: Mη(w;Q, τ)=(w−c)Q, (3.3.1) It will be useful to henceforth use the notation Q∗ j,η (w)and τ∗ j,η (w), j =S, H, to denote the optimal decisions by Health as a function of the selling price w, when maximizing (S)ocial welfare, or (H)ealth’s utility function. Also, observe the following definitions adapted from Chapter 2 adjusting for this contracting scenario, as a function of the endogenously set w: ¯ Qη τ(w)= max Q Q≤(Bh(τ)−δ+g)A(Q, τ)−gλ w−δ;Q > 0, Qη τ(w)= max Q P(Q;λF(τ)) ≥w−δ Bh(τ)−δ+g. Since Health’s problem remains unchanged with respect to Chapter 2, Pharma’s optimal 98 Theorem 4: a) When qhexists, then: a1) If wη 1exists, then τ∗ H,η = 1. a2) If wη 1and wη 2do not exist and Hη(Qη 2 ( ˜w),2; ˜w)≤0, then τ∗ H,η =       1 if c > ¯wη 2,H Q∗ H,η ( ¯wη 2,H )−¯wη 1,H Q∗ H,η ( ¯wη 1,H ) Q∗ H,η ( ¯wη 2,H )−Q∗ H,η ( ¯wη 2,H ) 2 otherwise a3) If wη 1does not exist, and wη 2exists, and ˜wQη 2 ( ˜w)>Γ, then τ∗ H,η =       1 if c > wη 2Q∗ H,η (wη 2)−¯wη 1,H Q∗ H,η ( ¯wη 1,H ) Q∗ H,η (wη 2)−Q∗ H,η ( ¯wη 1,H ) 2 otherwise a4) Else, then τ∗ H,η =       1 if c > ˜wQ∗ H,η ( ˜w)−¯wη 1,H Q∗ H,η ¯w1,H Q∗ H,η ( ˜w)−Q∗ H,η (wη 1) 2 otherwise . b) When qhdoes not exist, then τ∗ H,η = 1. The conclusions from Theorem 4 are very interesting. On one hand, and as anticipated, it is more likely that access will be restricted when Health maximizes his utility function versus when he maximizes social welfare. But on the other hand, we note that if Pharma selects price ˜w, then Health’s expected utility function will be positive and even the expected social welfare utility may be higher than when maximizing the latter was Health’s objective. The reason is that the utility maximizing formulation creates an artificial incentive compatibility constraint in order for Health to prefer the full versus restricted access level, preventing Pharma from extracting all the surplus, achieving a lower selling price compared to social welfare maximization, and resulting in a larger order quantity and possibly larger expected social welfare. Some situations where the latter could occur include a) when the maximum allowable budget is large, the relative size of the patient population with lower health benefits is relatively large and the difference in expected health benefits between the two patient populations is low; or b) when capacity building and manufacturing costs are 105 low, so that Pharma finds a sufficient incentive in the incremental revenue from higher access levels. 3.4 Exogenous price contracts with capacity buffer allowed So far we have assumed that Pharma has no excess capacity, and therefore Health may only order once per period. In this section we relax that assumption and we allow Pharma to build a capacity buffer, K, above Health’s order quantity Q. In such a situation, when demand exceeds Q, then Health can purchase up to Kadditional units, paying the per unit selling price wto Pharma, and incurring a per unit penalty cost p > 0, which is interpreted as a penalty for increasing the initial order size or for delaying the patient’s treatment. It is assumed that Pharma incurs an incremental cost pfor delivering units above Q, and therefore is indifferent between selling a unit of the drug during the initial order, or at a later point in time. The rest of the parameters are consistent with Chapter 2. Health has a per unit salvage value δfor purchased units in excess of the realized demand, and when total capacity, defined KT,(Q+K), is exceeded by demand, a per unit goodwill cost gis incurred. The transfer payment from Health to Pharma is redefined as T(w, Q, K) = wQ + (w+p)(min(K , (D(λ, F(τ)) −Q)+)). Pharma has no salvage value for unsold units. Additionally, we make a weak assumption to guarantee that risklessly purchasing a drug for an incoming average patient is superior to incurring the goodwill cost of not meeting that 106 patient’s demand. Mathematically, this is expressed as: Bh(τ) + g−w−p≥0.(3.4.1) Intuitively, by purchasing a drug from the capacity buffer and administering it to the patient, Health’s (possibly negative) margin is (Bh(τ)−w−p), to which we add the ’saved’ goodwill cost g. Next we formulate the exogenous price contract when Pharma is willing to incur part of the inventory risk by being able to build excess inventory. We use the symbol ’κ’ to denote exogenous price contracts with a positive capacity buffer allowed. Pharma’s expected profit function is: Mκ(K;Q, τ)=(w−c)Q+w(A(Q+K, τ)−Q)+−cK , (3.4.2) which reduces to Mκ(K;Q, τ) = (w−c)(Q) when K= 0. The social welfare’s expected utility function from Health’s perspective is: Sκ h(Q, τ;K) = (Bh(τ) + g)A(Q+K, τ) + δ(Q−A(Q, τ)) −gλF(τ) (3.4.3) and Health’s expected utility function is: Hκ(Q, τ;K) = (Bh(τ) + g−w−p)A(Q+K, τ)+(p+w−δ)A(Q, τ) −(w−δ)Q−gλF(τ) (3.4.4) To explain the formulation, notice that the salvage value δis only relevant for the first Qunits. The health benefits and the goodwill costs only depend on KT. The penalty cost p is only expected to be incurred for the difference between the expected administered drugs when KTversus Qunits are available. And the selling price wis deterministically incurred 107 for the first Qunits, and is expected to be incurred for the difference between the expected administered drugs when KTversus Qunits are available. For j=S, H, when the selling price is exogenously determined and Pharma is allowed to build a capacity buffer, define K∗ j,κ (Q,τ)as Pharma’s optimal capacity buffer for Health’s choice of Qand τ; and Q∗ j,κ (K)and τ∗ j,κ (K)as Health’s optimal order quantity and prescription policy threshold given Pharma’s choice of capacity buffer K. We begin by solving Pharma’s problem for any Qand τ, i.e., K∗ j,κ (Q,τ), j =S, H, and then proceed to find the equilibrium solution (or in some cases, solutions). Pharma’s Problem In order to obtain a more intuitive characterization of the solution, we initially show some necessary conditions for K∗ j,κ (Q,τ)>0 and then find upper bounds on the feasible quantity that Health may purchase. Lemma 10: For j=S, H,Q∗ j,χ < QΓis a necessary condition for Mκ(K∗ j,κ;Q∗ j,κ, τ∗ j,κ)> Mχ(Q∗ j,χ, τ∗ j,χ). Lemma 10 marks an incentive compatibility constraint for Pharma, since it is only optimal to build a positive buffer if it leads to an increase in the expected profits relative to the exogenous price-only contract presented in Chapter 2. Therefore, in the rest of §3.4 we will assume that the condition from Lemma 10 is satisfied. Lemma 11: Define ¯ KT(τ)= arg max Kint {wA(K, τ)−cK}to be the upper bound on Pharma’s optimal total capacity. For a given access level τ,¯ KT(τ)= max KP(K, λF(τ)) >c w, and ¯ K(Q,τ),¯ KT(τ)−Q. 108 The notation ¯ K(Q,τ)defines Pharma’s upper bound for the optimal capacity buffer as a function of the order quantity Qand access level τ. In other words, it shows the maximum level of inventory risk that Pharma is willing to accept by comparing the expected revenue of increasing the capacity buffer by 1 unit versus the corresponding (constant) deterministic cost c. While Lemma 12 gives the total capacity that Pharma would build in an unconstrained setting, two situations may occur. The first one is that Health chooses an order quantity larger than ¯ KT(τ)when Pharma’s relative understocking costs are lower than those of Health; this will be formally shown when we solve Health’s problem under each decision making criteria. The second situation is when Health’s constraints do not justify building ¯ KT(τ), as is expressed below Lemma 12: For any positive Qand K, let w(Q+K) + pK, be the largest possible realized expenses for Health. a) KΓ (Q)= max nbKcK≤Γ−wQ w+pois the largest capacity buffer that Health will be able to utilize given the budget constraint Γ and Pharma’s order quantity choice Q; b)∂((Γ−wQ)/(w+p)) ∂Q =−w w+p∈(−1,0); c) ∆ ,dw+p pe, is Health’s minimum order quantity increase to trigger a 1 unit increase in the total quantity that satisfies the budget constraint. Understanding the intuition from Lemma 12 is crucial to our analysis. First, it sets an upper bound on the feasible capacity buffer as a function of Q. Secondly, part b) explains that for an increase of 1 unit in Q,KΓ (Q)decreases in no more than 1 unit. This implies that the total capacity, KT=K+Q, that satisfies the budget constraint is non-decreasing in Q; in fact increases in 1 unit for every ∆ units that Qincreases, as is explained in Lemma 12c. As such, KΓ (Q)represents Health’s participation constraint with respect to its total budget. While for a fixed KTlarger values of Qare more likely to satisfy the budget constraint, the balancing effect is introduced in Lemma 13. 109 Lemma 13: KE(Q,τ)= min ndKeA(Q+K, τ)≥(w−δ)Q+gλF (τ)−(w+p−δ)A(Q,τ) Bh(τ)−w−p+gois the smallest capacity buffer that Pharma would need to provide in order for Health’s cost-effectiveness constraint to be satisfied. Lemma 13, while not a direct counterpart for Lemma 12, does contribute to a balancing effect in Health’s choices. One way to interpret it is that as Qincreases, the increase in Health’s expected utility function (both from the increased health benefits and from the decreased goodwill costs) of having the capacity buffer must be sufficiently large to overcome the escalating expected overstocking costs. As Qcontinues to increase, the benefits achieved by the buffer may not be sufficient due to the low probability of high values of realized demand, or because the increase required in the buffer is larger than what the budget allows. Integrating the results from this subsection, Proposition 9 provides Pharma’s best response for given Qand τ. Proposition 9: For j=S, H, assume Q∗ j,χ < QΓ. a) 0 < KE(Q,τ)≤min( ¯ K(Q,τ), KΓ (Q)), is a necessary condition for K∗ j,κ >0. b) KE(Q∗ j,χ,τ∗ j,χ)≤min( ¯ K(Q∗ j,χ,τ∗ j,χ), KΓ (Q∗ j,χ)), is a sufficient condition for K∗ j,κ >0. c) For given Qand τ, Pharma’s best response function when K∗ j,κ >0, is delimited by: KE(Q,τ)≤K∗ j,κ (Q,τ)= min( ¯ K(Q,τ), KΓ (Q)). Health’s Problem We now move to the analysis of Health’s problem. Before finding Health’s best response strategy under each of his decision-making criteria, we need to derive some additional results. In this respect, Lemma 14 finds the minimum order quantity necessary for Health’s budget to not become a restriction on Pharma’s desired capacity buffer based on her own critical fractile. Lemma 15 gives a sufficient condition for Health’s budget to prevent Pharma’s desired capacity buffer from being built. 110 Lemma 14: ˘ Qκ τ= min ndQePjΓ+pQ w+pk;λF(τ)<c w;Q < Γ wo, is Health’s minimum order quantity required for KΓ (Q)≥¯ K(Q,τ). Lemma 15: For j=S, H, if PΓ w;λF(τ)>c w, then K∗ j,κ (Q,τ)≤KΓ (Q)<¯ K(Q,τ). In other words, the two latter Lemmas provide a reference point for determining whether the budget constraint will be binding or not. On one hand, if the budget is large, the price is low, or Pharma’s overstocking cost is relatively large, then Pharma’s budget unconstrained maximization solution will yield the total capacity available. On the other hand, if the budget is low, the selling price and/or the penalty are high, or Pharma’s incentive to overstock is high, then Health’s budget will restrict the capacity built and the resulting drug amount available in the system. 3.4.1 Case 1κ: Maximizing expected social welfare In this subsection we solve the access and service level decisions when Health’s objective is to maximize expected social welfare, and Pharma is allowed to create an excess capacity buffer. Recall that for K∗ S,κ >0, it must be that Q∗ S,χ ∈ { ¯ Qχ 1,¯ Qχ 2}; otherwise Pharma has no incentive to provide the buffer. Consequently, Health’s problem is expressed as follows: max (Q,τ)(Bh(τ) + g)A(Q+K∗ S,κ (Q,τ), τ) + δ(Q−A(Q, τ)) −gλF(τ) subject to: KE(Q,τ)≤K∗ S,κ (Q,τ)= min( ¯ K(Q,τ), KΓ (Q)) T(w, Q, K∗ S,κ (Q,τ))≤Γ (Bh(τ) + g−w−p)A(Q+K∗ S,κ (Q,τ), τ)+(w+p−δ)A(Q, τ) −(w−δ)Q−gλF(τ)≥0 (3.4.5) 111 Notice that the objective function is increasing in the total quantity of drugs available, which implies that Pharma’s and Health’s objectives are aligned in the same direction, even though their participation constraints are in general different. In other words, both players benefit from higher levels of available inventory (given the feasibility constraints). Therefore we begin our formal analysis by using ¯ Qχ τas a reference point. Since we have assumed Bh(τ) + g > w +p, then purchasing any order quantity Kex-post satisfies the cost-effectiveness constraint, and we only need to check the absolute budget constraint for feasibility. As was mentioned above, there exists the possibility that Pharma’s optimal capacity buffer will be zero. Proposition 10 provides such situations. Proposition 10: Set τ∗ S,κ =τ. There are three scenarios that will result in K∗ S,κ = 0. i) If ¯ Qχ τ≥Qχ Γ. ii) If Γ−w−p w<¯ Qχ τ< Qχ Γ. iii) If ¯ KT(τ)<¯ Qχ τ< Qχ Γ. Proposition 10 allows us to further characterize the space for which the capacity buffer option is relevant. Conditions i) and ii) are related to the absolute budget constraint, so that any feasible combination of an initial order quantity and a positive capacity buffer will decrease social welfare when compared to the solution under the exogenous price-only contract. Condition iii) shows the case where Health is willing to accept a higher inventory risk than Pharma. This situation is more likely to occur as the selling price is relatively close to the production cost and relatively far from the benefit Bh(τ), or when the goodwill cost gis high for Health. Next, we define limits on the feasible order quantity considering the possibility that Pharma keeps an excess inventory stock. 112 Definition 3: a) Let ¯ ¯ Qκ τ= max bQc(w−δ)Q−(w+p−δ)A(Q, τ)<(Bh(τ) + g−w−p)A(¯ KT(τ), τ) −(Bh(τ)−δ+g)A(¯ Qχ τ, τ)+(w−δ)¯ Qχ τ, be Health’s largest order quantity that satisfies the cost-effectiveness constraint for a fixed total capacity ¯ KT(τ). b) Let ¯ Qκ τ= max bQc(w−δ)Q−(w+p−δ)A(Q, τ)<(Bh(τ) + g−w−p)A(Q+KΓ (Q,τ), τ) −(Bh(τ)−δ+g)A(¯ Qχ τ, τ)+(w−δ)¯ Qχ τ, be Health’s largest order quantity that satisfies both the cost-effectiveness constraint and the budget constraint. Definition 3a provides the largest order quantity that Health is able to purchase in advance, conditioning on the fact that Pharma will stock ¯ KT(τ)units. It is known that such order quantity will be at least ¯ Qχ τ; the larger ¯ ¯ Qκ τis, the higher the probabilities that the budget constraint will not be binding. In a similar vein, Definition 3b defines the largest order quantity that Health is able to purchase without violating the constraints. Note that on one hand, such quantity may be larger than Pharma’s choice of ¯ KT(τ), in which case Pharma would not build any excess stock. On the other hand, it is also possible that ¯ Qκ τ<¯ KT(τ), which would imply that purchasing ¯ KT(τ)is not feasible, and therefore the budget constraint will limit the total inventory available in the system. Lemma 16 incorporates the previous results to explain the behavior of the expected social welfare function, and Proposition 11 finds the optimal order quantity and capacity buffer for a given access level. Lemma 16: a) For a fixed τand KT,Sκ h(Q, τ;K) is weakly increasing in Q. b) For a fixed τand Q,Sκ h(Q, τ;K) is increasing in K. c) Assume ˘ Qκ τ≤¯ ¯ Qκ τ. Then, for fixed τ,Sκ h(Q, τ;K) is non-decreasing in Qfor Q≤¯ ¯ Qκ τ. d) Assume ˘ Qκ τ>¯ ¯ Qκ τ. Then, for fixed τ,Sκ h(Q, τ;K) is increasing in Qfor Q≤¯ Qκ τ. e) Assume ˘ Qκ τdoes not exist. Then, for fixed τ,Sκ h(Q, τ;K) is increasing in Qfor Q≤¯ Qκ τ. 113 Proposition 11: Fix τ∗ S,κ =τ. a) If ˘ Qκ τexists and ˘ Qκ τ<¯ Qκ τ, then: Q∗ S,κ =       ¯ ¯ Qκ τif δ > 0 h˘ Qκ τ,¯ ¯ Qκ τiotherwise ; and K∗ S,κ =¯ K(Q,τ)≤KΓ (Q) b) Otherwise, Q∗ S,κ =¯ Qκ τand K∗ S,κ =KΓ (Q)<¯ K(Q,τ). It is interesting to note that when δ= 0, there are multiple equilibrium between Health’s order quantity and Pharma’s capacity buffer K. However, this does not represent a problem since Pharma will build the same total capacity regardless of which equilibrium realizes, i.e., since the total capacity for Pharma is constant, Pharma’s capacity building choice is independent of Health’s order quantity. The role of Health’s order quantity, Q, will be therefore to allocate demand risk. As Qincreases, Pharma’s risk decreases, Health’s risk of overstocking increases, and Health’s total expenditures may either decrease or increase because as the initial order quantity decreases, the number of units for which Health expects to pay the penalty pincreases. Also interesting is that when either δ > 0, or ˘ Qκ τ>¯ Qκ τ, or ˘ Qκ τdoes not exist, the equilibrium solution is unique for a given access level and the budget constraint is the limiting condition. To complete the analysis, we turn to the problem of finding the optimal access level. Proposition 12: When I= 2, let qκbe a positive order quantity such that Sκ h(qκ,1) ≥ Sκ h(qκ,2); and Sκ h(Q, 2) > Sκ h(Q, 1),∀Q > qκ. a) β2> δ is a necessary and sufficient condition for qκto exist, and if equation (2.3.3) is satisfied and qκexists, it is unique and given by equation (3.4.6). g β1−δ+g λ A(Q+K, 2)+b1−b2 b1−δ+g=A(Q+K, 2) −A(Q+K, 1) A(Q+K, 2)  1 1−θ (3.4.6) b) If β2< δ, then Sκ h(Q, 1) > Sκ h(Q, 2),∀Q > 0. 114 sale to a secondary market; to avoid trivial problems, assume δ < c < β. If D(λ)> Q, a per unit cost, g, is accrued to Health for each patient arrival which does not receive the drug treatment due to a stock-out. To keep integrality, we will use bxcand dxeas the floor and ceiling functions, respectively. Except where it has been otherwise specified, all players are assumed to hold symmetric information about functional forms and parameters. Define A(Q), E [min[Q , D(λ)]] to be the expected quantity of administered drug treatments; E [max[0 , Q −D(λ)]] = (Q−A(Q)), to be the expected leftovers for Health; and E [max[0 , D(λ)−Q]] = λ−A(Q), to be the expected quantity of understocked units of the drug at the end of the period. Also, let T(w, γ, r) denote Health’s transfer payment to Pharma as a function of the contract parameters. Pharma remains a profit maximizer, and Health’s priority may be to either maximize expected social welfare, or maximize his entire expected utility function (i.e., social welfare minus the transfer from Health to Pharma). However, we should observe that Health has no way of trusting Pharma. As a result we define Health’s objective function, regardless of his priority, to be the smallest of the expected outcomes when the drug’s performance is either as guaranteed by Pharma, or as originally assumed by Health. We denote this contract structure with the symbol ρ. The manufacturer’s expected profit is: Mρ(γ, r;Q) = (w−c)Q−rQP(m;λ)G(γm)−v(m),(3.5.1) where notice that P(m;λ)G(γm) represents the probability of rebate. 121 The social welfare expected utility function is: Sρ(Q;γ, r) = min                        Slow(Q;γ, r) = (β−δ+g)A(Q) + δQ −gλ, if the health benefits are low, Shigh(Q;γ, r) = (β+ (b−β)(γ)−δ+g)A(Q) + δQ −gλ, if the health benefits are high, Since Shigh(Q;γ, r)≥Slow(Q;γ, r), then: Sρ(Q;γ, r)=(β−δ+g)A(Q) + δQ −gλ Health’s expected utility function is: Hρ(Q;γ, r) = min                        Hlow(Q;γ, r) = (β−δ+g)A(Q)−(w−δ)Q−gλ +rQP(m;λ), if the health benefits are low, Hhigh(Q;γ, r) = (β+ (b−β)(γ)−δ+g)A(Q)−(w−δ)Q−gλ, if the health benefits are high, In addition, two types of constraints are included in our analysis: a budget constraint: wQ ≤Γ,(3.5.2) where Γ is an exogenous upper limit on Health’s expenses for the drug under analysis; and 122 the cost-effectiveness constraints: Hlow(Q;γ, r )≥0,(3.5.3) Hhigh(Q;γ, r )≥0 (3.5.4) which guarantee that the expected net benefits derived from the drug’s approval are above some minimum threshold. Next, we map Health’s optimal order quantity as a function of the contract’s design, and analyze how it is affected by the contract parameters. Then we solve for Pharma’s optimal contract. 3.5.1 Health’s Problem Case 1ρ: Maximizing expected social welfare In this subsection we adapt the previously obtained results from the exogenous price-only contract (or equivalently, from the integrated chain) to the performance-based contract when Health is maximizing the expected social welfare. In order to do so, we use the same mechanism that was derived in Chapter 2 to find the potential optimal quantities as a function of the contract parameters. First, Lemma 18 defines the possible optimal order quantities under social welfare maximization. Lemma 18: a) ¯ Qρ high = max nbQcQ≤(β+(b−β)γ−δ+g)A(Q)−gλ w−δ;Q≥0o. b) ¯ Qρ low = max nbQcQ≤(β−δ+g)A(Q)−gλ w−δ−rP (m;λ);Q≥0o. c) Q∗ S,ρ = min QΓ,¯ Qρ low ,¯ Qρ high. As was the case under the simple contracts presented earlier, at optimality either the budget constraint or the cost-effectiveness constraint will be binding (or very close to binding, 123 because of integrality). Notice that when Health maximizes social welfare, our assumption of Health maximizing the minimum of the the two possible outcomes is irrelevant; this is because Health will purchase as many drugs as his constraints allow it to. This marks the difference here versus the analysis in Chapter 2, since Health’s lack of trust in Pharma implies that the cost-effectiveness constraint must be satisfied both if the health benefits are as guaranteed by Pharma, and if the health-benefits are as Health had initially acknowledged. Recall that even when the latter occurs, the rebate is not guaranteed since demand must be at least mfor the contract to be called upon. As a result, either rmust be sufficiently large or m, which is exogenous, sufficiently small so that the largest feasible order quantity is increased in a significant way. The next results explains how the contract parameters affect the feasible region for Health’s optimal order quantity. Lemma 19: a) ¯ Qρ high is weakly increasing in γ; b) ¯ Qρ high is independent of r; c) ¯ Qρ low is independent of γ; d) ¯ Qρ low is weakly increasing in r. e) Q∗ S,ρ is weakly increasing in γand r. Lemma 19 shows that as the value of the contract parameters increases, the largest feasible order quantity weakly increases. Intuitively, in the low benefits scenario as rincreases and everything else is kept constant, then the expected lump sum transfer increases, which may allow for larger order quantities to be feasible. Similarly, in the high benefits scenario as γincreases and everything else is kept constant, the health benefit that Health expects to see for each unit of administered drugs increases; while for the same Qthe utility function will increase, such difference may be large enough to allow the purchase of a larger quantity. Case 2ρ: Maximizing Health’s expected utility In this subsection we adapt the previously obtained results to the performance-based contract when Health is maximizing his expected utility function. To do so, we follow the same 124 mechanism as above, taking into account that there will be a limiting order quantity for each combination of parameters. Lemma 20 defines the feasible optimal order quantities under Health’s expected utility maximization. Lemma 20: a) Qρ high = max nbQcP(Q;λ)>w−δ β+(b−β)γ−δ+go. b) Qρ low = max nbQcP(Q;λ)>w−δ−rP (m;λ) β−δ+go. c) Q∗ H,ρ = min QΓ, Qρ low , Qρ high. The first two parts of Lemma 20 yield the budget unconstrained order quantity that maximize Health’s total utility function under the high and low scenarios, respectively. Lemma 20c then provides a parallel result to that of Chapter 2, since the optimal order quantity for a given access level will be given by the smallest of QΓ, and the order quantity that maximizes Health’s objective function; the obvious difference is that by considering two possible scenarios, there are two (possibly overlapping) curves, each of them with its own maximum value. Lemma 21 explains how the order quantity that maximizes the curve under each scenario changes as a function of the contract parameters. Lemma 21: a) Qρ high increases in γ; b) Qρ high is independent of r; c) Qρ low is independent of γ; d) Qρ low is increasing in r; e) Q∗ H,ρ weakly increases in γand r. Now that Health’s decision making criteria has been established, we proceed to analyze Pharma’s problem of optimally designing the performance based contract. 125 3.5.2 Pharma’s Problem In this subsection we derive the optimal parameters of the performance-based contract from Pharma’s perspective. Since the following results apply for both of Health’s decision-making criteria, we use the letter j=S, H to denote the expressions in a consistent manner without being repetitive. Proposition 15: a) r∗ j,ρ (γ)=A(Q∗ j,ρ) Q∗ j,ρ  b−β P(m;λ)(γ); b) γ∗ j,ρ (r)=Q∗ j,ρ A(Q∗ j,ρ)P(m;λ) b−β1 r. Proposition 15 states the optimal relationship between the contract parameters in order to avoid offering Health unnecessarily benevolent conditions in the contract that can’t be compensated by larger order quantities. In other words, it implies that when the performancebased contract is implemented, then at optimality ¯ Qρ low =¯ Qρ high, under social welfare maximization; and Qρ low =Qρ high, under Health’s expect utility maximization. If that was not the situation, then one of the constraints would have a positive shadow price created by either a guarantee γ, that could be decreased (decreasing the probability of paying a rebate), or by a rebate rate rthat could be decreased (decreasing the value of the rebate in case the guarantee is not satisfied), in both cases without inducing a decrease in Health’s order quantity. Furthermore, this balancing relationship allows us to reformulate Pharma’s problem as that of a single decision variable as is expressed next. Lemma 22: At optimality, Pharma’s problem under a performance based contract may be rewritten as a single variable problem, and is given by equation (3.5.6): M(γ;Q∗ j,ρ) = (w−c)Q∗ j,ρ −A(Q∗ j,ρ)(b−β)(γ)G(γm) (3.5.5) Unfortunately we can’t obtain the explicit solution for the parameters, but Proposition 15 gives the conditions that need to be satisfied by the optimal guarantee factor γ, and 126 this can be used to obtain the optimal rebate rate r. Similarly, we are unable to determine analytically whether Pharma has an incentive to truthfully reveal her private information. Intuitively though, as mbecomes large, the sample mean will approach the true mean, and setting γ=πwould imply paying a rebate with approximately a 50% probability. As a result, we expect that under this contract’s structure, Pharma will tend to understate its private information, or not provide any information at all. The latter situation occurs when the condition from Corollary 3 is not satisfied. Corollary 3: Define Q∗ j,χ as Health’s optimal order quantity under exogenous price-only contract with parameter w. Pharma can benefit from the performance based contract only if: (w−c)Q∗ j,ρ −Q∗ j,χ Q∗ j,ρ > r∗ j,ρG(γ∗ j,ρm)P(m;λ) + v(m) Q∗ j,ρ . Corollary 3 gives the necessary condition for the performance based contract to take place. If this is not met, then Pharma can simply set a rebate rate equal to zero, and the system will behave as in an exogenous price-only contract scenario. 3.6 Conclusions Chapter has presented three mechanisms that the upstream player, Pharma, may use in order to increase her profits relative to the exogenous price-only contract presented in Chapter 2. These contracts are an enedogenous price-only contract; a capacity buffer under exogenous transfer price; and a performance-based contract under exogenous price. For the endogenous price-only contract, we find a very interesting situation. The manu127 facturer has an incentive to increase the price and decrease order quantity as long as total profits keep increasing, and therefore the cost-effectiveness constraint tends to be binding or very close to binding under social welfare maximization. In fact, when the optimal access is limited, both social welfare and expected utility maximization yield extremely similar, or even equal, results; Pharma extracts all of Health’s surplus. However, when the health payer maximizes his expected utility, the manufacturer is not able to induce full access and extract all of the health payer’s surplus simultaneously because of the threshold price mentioned earlier. This implies that if the manufacturer wishes her product to be considered available to a larger fraction of the patient population, then she must reduce the price, which under some parameter combinations, may even result in a larger order quantity than that obtained by social welfare maximization because of the incentive compatibility constraint imposed by comparing the objective function under the different access levels. This result is a potential argument for why some markets allow the manufacturers to set prices freely and then act as profit maximizing entities. In short, according to the model, utility maximizing may be in some cases a more efficient tool for achieving social welfare than social welfare maximization itself. After that, the buffer capacity contract was introduced to include Pharma’s willingness to adopt some of the inventory risk. However, we have kept our distance from the infinite inventory assumption by considering Health’s cost-effectiveness and budget constraints, which along with Pharma’s incentive compatibility constraint, determine the optimal capacity buffer. We find that this contract is most useful to the manufacturer when the budget constraint under price-only contracts is not binding and the manufacturer’s per unit overstocking/understocking cost ratio is much lower than that of the health payer. However, when Health’s budget is large, the buffer capacity contract will result in a lower initial order quantity by Health compared to the exogenous price-only contract, implying that Pharma’s certain revenue will be strictly lower if access level remains unchanged. For the health payer and the patients, the buffer capacity contract will be most useful when demand uncertainty 128 is high, goodwill cost is high, and the available budget is relatively large. Finally, we find that under social welfare maximization, the buffer capacity contract can’t decrease access level nor available inventory relative to the exogenous-price only contract; however when the health payer maximizes its expected utility, access level and inventory available can either decrease or increase. In the end, we have proposed a novel performance-based contract where the pharmaceutical manufacturer has the option to increase the health-payer’s willingness to pay for a given drug by offering a partial guarantee on the average realized health benefits. While other similar contracts have been studied and implemented, the particularities of the approach here developed is that given the manufacturer’s informational advantage, the health-payer must base its decisions on the possibility that every contracting scenario may realize. As a result, if Pharma sends a signal indicating a high level of trust on her originally announced health-benefit, then she will also need to offer Health a large rebate as collateral. From this perspective, and while we cannot prove it analytically, the manufacturer is not expected to send a reliability signal that is higher than her privately held knowledge when the variance of Pharma’s private information is high, because doing so would increase the probability that the rebate will need to be paid. In short, the health payer can only calculate his expected utility based on his own perception of the health benefits and the manufacturer’s signals, and bases his decisions on the scenario that imposes the tightest constraints; but the manufacturer has visibility on the probability that each scenario may occur so that if she were completely confident that her announced health benefit will be equal to the realized benefit, then she could set an extremely large rebate without any negative consequences. Another interesting property of this contract is its attempt to distance itself from the so-called costcontainment initiatives and become a truly risk sharing mechanism. For example, without the need to alter the initial selling price, both the manufacturer and the health payer benefit from a successful outcome: the payment to the manufacturer increases due to the possibility of charging the initially negotiated price and selling a larger than initially negotiated 129 order quantity, cost-effectiveness is maintained, and drug availability increases. In terms of the downside risks for the players, the manufacturer assumes most of the risk from lower than expected health benefits, while the payer assumes the risk relative to the size of the demand. It is worth discussing the implementability of this contract when multiple patient categories can be treated by the same drug, and explain why the access level decision was not included in this model. The situations where it is not expected to add value are when the budget surplus from Health is low so that the manufacturer’s risk acceptance can’t be rewarded with a larger revenue stream, and when the cost of verification is large. As for the exclusion of the access level decision, the main reason is that a fair assessment of the drug’s performance would require the proportion of patients from each category within the sample to be the same as in the population. This situation is on one part harder to control, and additionally would lay an additional layer of risk on the manufacturer’s utility function, as it would be dependent on the realized distribution of patient arrivals which is prone to being manipulated by an unethical health-payer. Instead, the belief is that this contract design should be implemented for a single category of patients per drug (even if, and perhaps specially when, the drug is administered to multiple categories), and that this category should be the one where the health-payer’s trust in the manufacturer’s announced health benefits is low and the manufacturer’s level of confidence is high. In such situations, the contracting mechanism can allow for increases in access and service level by increasing the range of order quantities for which cost-effectiveness is achieved, and in general increasing the health-payer’s expected utility function. Furthermore, since manufacturers tend to lose patients’ and prescribers’ goodwill following negative health outcomes, it is considered unlikely for this type of contracts to be viciously implemented. Last, it is true that a policy-maker or a pharmaceutical manufacturer may implement the contracts here analyzed in a simultaneous manner. Our analysis does not show any evidence against doing so. Rather, we have attempted to identify the virtues and shortcomings 130