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COMMON AGENCY THEORY AND THE INDUSTRIAL ORGANIZATION OF HEALTH CARE By WILLIAM EDWARD ENCINOSA, III A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1995 ACKNOWLEDGMENTS This work was supported financially by a Jim WalterLanzillotti Dissertation Fel lowship. I thank the Public Policy Research Center at the University of Florida for their generous support. I am indebted to David Sappington for his insightful comments, invaluable advice, and constant support in the difficult task of translating economic theory into public policy. I am also grateful to Tracy Lewis, Jon Hamilton, Mike Ryngaert, and Steve Slutsky for many helpful discussions. In particular, thank David Martimort of IDEI, France, for his help with common agency theory. I also thank Professor Louis Block of the Mathematics Department for providing me with very generous computer facilities. This work has benefited greatly from the comments of seminar participants at Harvard University's John Kennedy School of Government, Carnegie Mellon's Heinz School of Public Policy, and IUPUI. I am also grateful for the many helpful discussions with the faculty and scholars participating in 1995 Robert Wood Johnson Health Policy Research Programs at the University of CaliforniaBerkeley, the University of CaliforniaSan Francisco, and the University of Michigan. Finally, I am especially grateful to my parents for their continuous support and encouragement during this work. TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . .* . a a S ii ABSTRACT CHAPTERS GENERAL INTRODUCTION COST SHIFTING AND GOLDPLATING IN HEALTH CARE Introduction . . . The Model. . . The Overinvestment Problem Mixed Payment Systems Goldplating . . Public Policy Implications Conclusion . . . * S S . S S S S S . a a S S S S S S S . * 0 0 5 5 S S S S S S S S S S S S S S S * a . S S S S p a S . . * S S S . 0 0 0 S 0 S S S * p . 0 S S a S S OPTIMAL HOSPITAL CAPITAL STRUCTURE Introduction . The Model . Debt Strategies Conclusion . * S S p . . S S S 0 5 0 0 * . 5 S S 0 .*. S S S S S S . S S S P 0 S S S S S S S 0 S S S S S * a 5 S . . . P P 5 5 S S S PRICING UNDER EXCLUSIVE DEALING Introduction . . . Common Agency . The Monopoly Benchmark Competition for an Exclusive Hospital Exclusive Dealing Conclusion . . . * . S S 0 0 S S 5 S . * S S 0 0 S S 0 5 5 S . Dealer . 5 5 0 5 0 5 5 S S S S S S P 5 5 a a S S S P 5 . * 0 S S . S . 0 0 S S S 0 * S S P . S S S S S CONCLUDING REMARKS DTfl I D A DTXTrT A T cOTTTu APPIENDIX REFERENCES Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMMON AGENCY THEORY AND THE INDUSTRIAL ORGANIZATION OF HEALTH CARE William Edward Encinosa, III May 1995 Chairman: Professor David E. M. Sappington Major Department: Economics Common agency theory deals with a competitive version of principalagent theory in which multiple principals contract with the same agent. This dissertation develops common agency theory for applications to the industrial organization of the health care industry. In the first essay, the efficiency of three health care systems is examined. In a multipayer system the public payer (Medicare) uses a mix of prospective payments and passthrough payments, while the private payer (a managed care insurer) uses a qualitybased reimbursement rate through utilization review. Costshifting in the multipayer system induces the hospital to overinvest in technology, even under pri vate insurer utilization reviews. Furthermore, passthrough payments of capital are scaled back in equilibrium since they create a moral hazard problem that allows the hospital to goldplate, i.e, to invest in wasteful, nontechnological capital. This may explain Medicare's current policy to phaseout passthrough payments. In a single payer system investment in technology is curtailed, and goldplating can arise. When +t*I I nt*" I niarfl nannn o k niaa +b0IQ n\t'no nn A +0 n I ni ,wa A ry 4 ^ n nyi ljt fv^ The second essay studies the interrelationship between a hospital's capital struc ture and the payment plans designed by Medicare and the private insurance sector to reimburse the hospital's cost of capital and technology. To counter the opportunism of a managed care private insurance sector engaging in utilization review, the hospital will use debt when bankruptcy is costly. The final essay derives the equilibrium hospital contract with a physician group that has private information on the risk rating of the health maintenance organization it serves. Bundled physician charges are then derived when hospitals compete for an exclusive contract. CHAPTER 1 GENERAL INTRODUCTION The United States is presently undergoing a national health care crisis. In 1994, expenditures on health care exceeded $ 1 trillion, more than 14 percent of GDP. Med ical costs are projected to soar to 19 percent of GDP by the year 2000 (Clinton 1993). Per capital, the United States spent 40 percent more on health care than Canada, the second highest spender, and twice as much as the major European countries in 1991 (Starr 1994). 1990, United States was spending more on health care than on education and defense combined. Yet, paradoxically, about 18 percent of Ameri cans do not have medical insurance. In fact, 26 percent of Americans had no health insurance coverage at some time between 1987 1989 (Starr 1994). Moreover, 86 percent of these uninsured were members of working households (Roberts 1993). Among OECD countries, the United States ranks nineteenth in infant mortality and twentyfirst in life expectancy for men (Clinton 1993). level of medical expenditures, Thus, for its extremely high United States does not seem to be delivering a commensurate level of health care to its people. This crisis has lead to an unprecedented public policy debate over federal reform of the health care industry. The last major federal health reform to occur was in 1965 with the establishing of Medicare to insure the elderly and the severely disabled. At that time, health care was only 6 percent of GNP. Yet, now Medicare spending is running 23 percent higher than the rate of inflation (Clinton 1993). Consequently, the Medicare Trust Fund will be empty by the year 2002. The Reagan Adminis tration did make substantial changes in Medicare payments to hospitals with the introduction of the Prospective Payment System (PPS) in 1984. Under this system, hospitals receive a fixed payment for each patient in a specific diagnostic group, in 2 held Medicare reimbursement expenditures down initially. However, after 1989, hos pital expenditures began to soar again, accounting for about 40 percent of all medical expenditures in the United States. Parallel to this reform in the public sector, the private insurance sector and the hospital industry are undergoing major transformations. Over the past years, more than 400 hospitals have sought mergers. More importantly, there has been a significant movement to merge insurance and health care provision. These mergers have taken the form of health care maintenance organizations (HMOs) and preferred provider organizations (PPOs). Over half of the urban population is projected to be enrolled in HMOs by 1998. A key feature of these managed care facilities is that they engage in utilization review. That is, they monitor the appropriateness of medical treatment and review patient outcomes. This prevents doctors and hospitals from performing unnecessary, expensive procedures. Arnold Relman, former editor of The New England Journal of Medicine, estimates that roughly one third of health care expenditures are medically unnecessary. an initial one time savings of about 10 It appears that percent. HMOs have experienced Beyond that, managed care cost advantages seem to be waning. Thus, we see a failure in both the private and public sectors of the health care market to contain medical costs. The following three essays in this dissertation will take an industrial organizational approach to examining medical cost containment incentives under both regulation and competition. Industrial organization provides a set of tools to analyze these changes in markets and the consequences of public policy initiatives. we will focus on the the tool of common agency theory. In particular, This is the theory of multi principal agent theory, which investigates how two rival principals should design incentive contracts for an agent that they happen to share. frequently in the health care industry. This scenario arises quite For example, the first two essays consider 3 contrast, the third essay takes two rival hospitals to be the principals competing for the exclusive patronage of an HMO (the agent). Using this common agency approach, I am able to gain new important insights into the industrial organization of the health care industry. The first essay, titled CostShifting Goldplating in Health Care, examines the efficiency of three health care systems. First, I show that a singlepayer system under PPS (similar to Canada's health system), in which the government is the sole medical insurer, induces the hospital to underinvest in technology and to undersupply treatment quality. However, I show that this singlepayer system can be made efficient by implementing a mixed payment in which a costbased element is added to the PPS rate. Next, I examine a multipayer system in which part of the market is insured by Medicare (under PPS), while the remainder of the market is insured by a Managed Care private insurance sector (e.g., HMOs). This system induces the hospital to overinvest in technology to undersupply treatment quality. This reflects the current state of the United States' multipayer system. By using an inverse elasticity rule in its reimbursing of technology and operating costs, Medicare is able to shift operating costs to the private insurance sector. Even though the private insurer is allowed to engage in aggressive utilization reviews of the hospital, this is not enough to counter the distortion caused by Medicare's costshifting. in technology persists under utilization review. This result di in health care that managed care will mitigate Medicare's cos Thus, overinvestment spels a common notion tshifting and bring the multipayer system back to efficiency. Another commonly held belief is the idea that a mixed payment system (PPS with a costbased element) will correct the distortions in the multipayer system as it did in the singlepayer system. In fact, I show the exact opposite occurs. Mixed   4 the multipayer system. In essence, the mixed payment just gives Medicare another instrument to shift costs to the private insurance sector. Moreover , I show that the multipayer system induces the hospital to goldplate, , to invest in wasteful capital such as fancy artwork and lobby waterfalls. To deter goldplating, Medicare must scale back its passthrough payment of capital. This may partially explain Medicare's 1991 decision to phaseout passthrough payments of capital. Finally, I introduce the allpayer system in which the private and public insurance sectors trilaterally negotiate with the hospital over a uniform reimbursement rate. By instituting a uniform rate structure, this allpayer system prevents Medicare from costshifting. Moreover, by dismantling the passthrough payment, this system is immunized against goldplating. The second essay, Optimal version of the model in the first Hospital essay. Capital Structure, introduces a stochastic This enables me to examine how forprofit hospitals should finance their investments under the risk arising from volatile nature of hospital admissions. highly I show that the hospital will issue some debt in order to counter the private insurer's opportunistic behavior in utilization reviews. By using debt, the hospital purposely exposes itself to a positive risk of bankruptcy. Since bankruptcy proceedings are costly, this forces the private insurer to increase its reimbursement rate in order to reduce the risk of bankruptcy. the hospital outweighs the risk of using debt. This rate increase for Thus, I predict that even as forprofit hospitals lose their taxshelter advantage with debt, hospitals will continue to issue bonds in order to mitigate the everincreasing practice of utilization review by the managed care sector. In the last essay, titled Pricing under Exclusive Dealing, I examine how hospital competition for an exclusive contract with a physician group affects the physician rourn's nririnu" of service to an lIMO. This rnommnn a.pancv model is much more 5 information. Specifically, the physician group (the agent) has private information on the HMO's demand for health care. The hospitals (the principals) do not know the HMO's demand, and so must design incentive contracts to induce the physicians to reveal this information. Each hospital offers a reimbursement rate to the physician group for the use of the hospital's technology. The physician group than bundles this hospital rate with a physician reimbursement rate and charges this price bundle to the HMO. I examine how hospital competition and integration affects the physicianinsurer bargaining over prices and utilization levels. I derive the equilibrium hospital con tracts for two rival hospitals competing through the same shared physician group which is privately informed about the insurer's bargaining power. For substitute (complementary) medical procedures, competition through a common physician group results in higher (lower) physician prices and less (more) severe underutilization of medical services when compared with a multispecialty hospital merger. Under strong substitutes, the hospitals instead compete headtohead for the exclusive services of the physician group. For an insurer with a weak (strong) bargaining position, ex elusive dealing results in lower (higher) physician prices and underutilization (over utilization) when compared with a hospital vertically integrated with the physician group. CHAPTER 2 COST SHIFTING AND GOLDPLATING IN HEALTH CARE Introduction Health care reform has been one of the most hotly debated public policy issues during the Clinton Administration. It is widely agreed that any health care re form must first address cost containment. In 1994, expenditures on health care in the U.S. are expected to exceed $1 trillion, more than 14 percent of GDP. Medical costs are projected to soar to 32 percent of GDP by 2020. Many health care economists (Evans (1986), Newhouse (1988), and Weisbrod (1991), for example) argue that the present system of financing hightech medical technology is responsible for the rapid explosion in medical costs. According to the former longtime editor of the New England Journal of Medicine, Arnold Relman, technology is the "engine behind the rise in medical costs," driven by the excessive number of doctors "trained to provide hightech, expensive services" (Relman (1989)). In the debate over how to deal with this overinvestment in technology, controversy often erupts over the merits of three different health care systems: the allpayer system. the singlepayer system, the multipayer system, and In this paper we offer an economic analysis of these three tems of health care and provide answers to the following key questions: (1) Which of the three systems best provides the hospital with the incentive to invest in the so cially efficient level of technology? (2) What is the role of price negotiations in these systems? (3) How do the incentives of costbased reimbursement, prospective re imbursement, mixed reimbursement, and qualitybased reimbursement differ among the three systems? (4) To what extent should the government make passthrough payments of hospital capital in each system? and (5) Which system is susceptible to 7 The United States' health care system is currently a multipayer system, in which the public and private insurers set reimbursement rates noncooperatively. A com mon criticism of multipayer systems is that the public insurer (Medicare, Medicaid) is able to shift costs to the private insurance sector. While such cost shifting has long been criticized for being unfair or inequitable for patients, it is only recently that costshifting has been shown to lead to economic inefficiencies such as decreased pa tient hospital length of stay (Glazer and McGuire (1994)) and possible technological underinvestment (Ma and McGuire (1993)). However, it has yet to be demonstrated how costshifting may induce excessive investment in hightech medical equipment. The goal of this paper is to link technological overinvestment to costshifting. This explanation of overinvestment is in contrast to the traditional "medical arms race" (MAR) explanation in which it is argued that hospitals compete by providing too many hightech medical services. The implication is that capital is wasted, leading to higher costs without commensurate benefits (Robinson and Luft (1985), Kopit and McCann (1988), and McManis (1990)). This seems to be the prevailing explanation for hospital overinvestment. In fact, antitrust judges have even embraced this idea to the extent of permitting hospital mergers on the grounds that it would bring the med ical arms race to an end 1 Over the last five years, more than 400 hospitals sought mergers2. Nevertheless, hospital costs continue to escalate, suggesting that incentives for overinvestment may not be due entirely to competition among hospitals3. This paper reveals and analyzes a more fundamental source of the overinvestment prob lem. We show that excessive hospital investment can result from the costshifting induced by competition among payers, even in the absence of any competition among hospitals. 1For example, in a recent decision to permit a merger between the two largest hospitals in Roanoke, Virginia, the district court judge wrote,"As a general rule, the hospital rates are lower, the fewer the number of hospitals in an area"(United States v. Carilion Health System 892 F2d Some argue that the government's costshifting in a multipayer system may be counteracted by the rapid upsurge of "Managed Care" plans in the private sector (such as health maintenance organizations preferred provider organizations). However, a key finding here is that hospital overinvestment in technology persists even in such a managed care environment, even if private insurers can set aggres sive qualitybased reimbursement rates through utilization review. That is, after a utilization review of the hospital's quality and technology, the private insurer can update its reimbursement rate. This reflects the current "Outcomes Movement," which Relman (1988) dubbed the "third revolution in medical care." The basic goal of the outcomes movement is to link reimbursement rates data on patient outcomes and hospital quality. An interesting result of our paper is that these qualitybased, or outcomesbased, hospital rates cannot resolve the overinvestment problem of the multipayer system. That is, these aggressive payments do not provide strong enough incentives to counter the costshifting problem. The reason for this conclusion is that with a firstmover advantage, the government can impose a costshifting externality on the bargaining process between the hospital and the private payer by using a com bination of two payments: (1) a prospective payment, which is a fixed payment per patient, and (2) a passthrough payment which pays for a percentage of capital and technology. To compensate for this externality, the hospital must always overinvest in technology in order to improve its bargaining position with the private insurance sector. Interestingly, Medicare used such a combination of passthrough payments and prospective payments up until 1991, when it decided to phaseout passthrough pay ments over a ten year period. Why would Medicare phaseout passthrough payments, since doing so would seem to diminish its ability to shift costs to the other payer? To answer this, we show that the passthrough payment is susceptible to hospital for doctors, etc. While the investment in hightech medical equipment is excessive, it is still a productive asset in that it has value to patients. Goldplating, on the other hand, is completely wasteful in that it has no value to patients. To deter gold plating, Medicare must scale back its passthrough payments, tradingoff diminished costshifting for less goldplating. However, overinvestment in technology as well as some goldplating may still persist in the multipayer equilibrium. How might these persistent problems with the multipayer system be solved? A popular idea is to replace the multipayer system with a singlepayer system, in which the government is the only medical insurer4 While such a change would eliminate costshifting, it would also, however, severely curtail socially beneficial investments in technology. A large strand of the literature has recently suggested that the underivestment problem of prospective payments can be prevented by mixing the prospective payment with a costbased retrospective payment. In contrast, we show that this restoration of efficiency will not occur in reality since the costbased element of the mixed payment is susceptible to goldplating. As a result, goldplating as well as underinvestment in technology persist in the singlepayer equilibrium. To solve the problems of the multipayer and singlepayer systems,we introduce an allpayer system. Here, the public payer, the hospital, trilaterally negotiate a uniform reimbursement rate which can quality. However d the private payer vary with delivered Allpayer systems are used in Japan, Europe, and in four American states. , these do not typically use qualitybased reimbursement. This is unique to our allpayer model. The key result of our paper is that this particularly aggressive allpayer system is technologically efficient. Moreover, this allpayer system is immune to costshifting and hospital goldplating. This chapter is organized as follows. In Section 2 , the three health care systems are introduced. In Section 3, the equilibrium reimbursement plans and investment 10 levels under the three health care systems are derived. Mixed payments are analyzed in Section 4. Goldplating is then introduced in Section 5. Finally, in Section 6, the public policy implications of the analysis for medical cost containment are discussed. All proofs are relegated to the Appendix. The Model First, we set up a model of a multipayer health care system. To begin, we consider the demand side of the health care market. For simplicity, we assume that consumers in the health care market are fully insured5. Thus, market demand is assumed to be price inelastic with respect to the price of medical services. However, following Rogerson (1994) and Ma (1994), we assume that the hospital's demand for admissions X(I) is directly influenced by the quality of care or intensity of care offered by the hospital. ProPAC's (1993) working definition of 'intensity' is the number and complexity of patient care resources, or intermediate outputs, used in producing a patient care service. Intensity I may include length of stay, services per admission, special amenities, etc. Unlike Ma and McGuire (1993), is a function of the hospital's we do not assume that that hospital level of technology volume T. Recent research (e.g., Dranove and White (1992)) suggests that technology usually does not attract patients to the hospital directly, but instead attracts physicians to the hospital. This results in a physiciandriven 'medical arms race' (MAR) in which hospitals overinvest in high tech equipment just to attract the best doctors. This phenomenon is the traditional explanation for excessive investment in hospital technology. However, recent empiri cal evidence indicates that the MAR effect has diminished or disappeared altogether (see Zwanziger and Melnick (1993)). Thus, it remains to determine what else besides 5We assume that patients obtain insurance from only one insurer at a time. For a discussion a MAR may be driving the overinvestment in medical technology. A central finding of this paper is that payer competition can cause overinvestment in technology, even in the absence of a MAR, and even under prospective payments. In order to investigate payer competition in isolation, we first remove the incen tive for a MAR by assuming that the hospital's number of physicians with admitting privileges is fixed exogenously. Thus, the hospital will have no demandside moti vation for overcapitalization through technology (however, demand will be induced by hospital intensity I). Instead, the hospital's only incentive for overinvesting in technology will be shown to arise from the supplyside effects of payer competition. On the supply side, the hospital has an operating cost of c(I,T) per patient. There are no fixed costs associated with supplying I. The hospital's only fixed cost is T. Following Ma and McGuire (1993), n will be the fraction of patients that are insured by the public payer (Medicare, Medicaid); the other (1n) of the market is insured by the private payer6 . The public payer pays a reimbursement rate a per patient to cover c(I,T) and to help pay for T. In addition, the public payer can make a pass through payment pT that is independent of the number of discharges, where pE [0, 1]. This payment plan represents the Prospective Payment System (PPS) administered currently by Medicare and by some state Medicaid programs (ProPAC 1993). The private payer will pay a rate / per discharge. f is determined by negotiations between the private payer and the hospital. Historically, passthrough payments have never been instituted by private payers, and so will not be considered here. Our model of the health care system focuses on the interactions among three actors: the public payer, the hospital, and the private payer7 The public payer and the private payer will compete in setting their rates (a, p) and /, respectively, in order to reimburse the hospital's operating cost c(I,T) and cost of technology T. However, 6We are essentially modeling the multipayer system as a von Stackelberg common agency, where the insurers are the principals and the hospital is the agent. Our model departs from the usual 12 this payer competition will further entail opportunistic behavior on the part of each payer. First , we describe how the public payer might be able to shift some of the costs of treating Medicare patients on to the private payer. A well known problem in health care is that public insurers, such as Medicare and Medicaid, often do not reimburse the hospital for the fair share of their patient unpaid costs to the private insurance sector8. shift costs in this manner, ts' costs, thereby shifting a burden of To model the public payer's ability to we give the public payer the firstmover advantage as a price setter. Thus, in stage 1 of the fourstage game, the public payer (Medicare) sets the margin a and the passthrough payment parameter p. Medicare is presumed unable to update its payment plan after the hospital selects I and T, perhaps due to a lack of managerial resources9. Anticipating the results of the next two stages of the game, the public payer will select a reimbursement plan (a, p) in order to maximize the net social benefit to the segment of the market it serves: M(a, p) = n[U(I, T) aX(I)] pT, where U(I,T) is the gross social benefit when the hospital provides technology level T and treatment intensity level I. Note that an incomplete contracts approach is adopted. We assume that the insurers cannot write complete ez ante contracts con tingent on the hospital's technology and treatment intensity. Stateoftheart inno ovation in medical technology and medical procedures often advance so rapidly that it is usually difficult, if not impossible, for the private insurance sector to characterize precisely all relevant aspects of a hospital's technology in a contract. In stage 2, anticipating the per patient fee that the private payer will negotiate (f) in stage 3, the hospital chooses the level of investment T and intensity I in order sThe American Hospital Association estimates that Medicaid payments covered only about 80% of average costs in 1991. Hospital profit margins on Medicare's PPS patients averaged 4% in 1991 (AHA 1992). to maximize profits:'0 Y(I,T) = [na + (1 n)/3]X(I) c(I,T) + pT While public payer costshifting has been a major focus of recent research, oppor tunism on the part of private payers has been neglected in the literature. The salient feature of this paper is that it introduces two forms of private payer opportunism that have recently emerged with the rapid upsurge of "Managed Care" as preferred provider organizations (HMOs))" plans (such (PPOs) and health maintenance organizations A key feature of these managed care plans is that they often engage in utilization review. That is, they monitor the hospital's choice of quality I and level of technology T. This gives the private payer the ability and flexibility to quickly update rates in response to the hospital's behavior. Thus, the aggressive monitoring of utilization review allows the private payer to behave opportunistically by basing its rate p on the observed I and T. In ore tween the hospital and the private payer, der to avoid a complex dynamic game be we assume that the hospital's choice of I and T in Stage 2 is immutable. One possible justification for the hospital to commit to T and I is that it may be very costly for the hospital to constantly adjust T and I in a dynamic pricing game. For example, to induce the full level of demand X(I) may require a period of commitment to the level I. That is, the doctors must build a relationship with their patients over a period of time in which they are credibly com mitted to a guaranteed level of care I. In addition, it may be costly for the hospital to quickly purchase or resell hightech equipment on the spot market. Like the prospective payment a, the payment /f is a fixed rate per patient. ever, unlike a, the payment B may depend explicitly on I and T How Thus, to distinguish p from a prospective payment, we will call f/ a qualitybased reimbursement rate. toWe assume that the hospital cannot discriminate by payer in their intensity provision. o . ,.. InA\A\ r ..... . J. i : C.^ ..I .. .. We assume the private payer has bargaining power vis a vis the hospital. Such power may arise in practice when, for example, the private insurer forms large net works of doctors and patient s across several markets. To capture the private payer's purchasing power, f/ is assumed to be determined in stage 3 as the solution to a generalized Nash bargaining game12 . Presuming the private insurance sector to be competitive, the private insurer's objective is to maximize the net expected social benefit to the patients it represents: V(3) = (1 n)[U(I, T) X(I)]. Note that if the hospital leaves the bargaining table, it derives income only from serving the patients insured by the government. That is , the hospital's threat point is endogenously determined by the public insurer's reimbursement plan. However, we assume that the investment T is not hospitalspecific and is completely reversible and redeployable. This is characteristic of hospital technology. In such a case, Encinosa (1994) shows that private owners of the hospital will always redeploy the hospital's capital to other markets if the hospital makes the outofequilibrium decision not negotiate with the private insurance sector. Consequently, the hospital's threat point will always be zero. any outside hospital. Sin Next, the private insurer has no purchasing power with ce the private payer cannot obtain a discount elsewhere, we will normalize the private insurer's threat point to 0. With a (0,0) threat point, the generalized Nash bargaining solution f/* maximizes Y'V1i while guaranteeing 0, where q E [o,1 is the relative bargaining power of the hospital. In the final stage 4, demand (I) is realized (no rationing is allowed). The public insurer pays the hospital aX(I)+pT , while the private insurer reimburses the hospital px(i). When n E (0, 1), a multipayer system emerges in which the public and private r~~~CA ... I .La: a a: D. 1. .. nL a a1 I. a~ 4A a~l nII ef a a ana4 1..n ta IxTE  1, ,.,, rL, i 15 singlepayer system, in which the government is the sole insurer and sets hospital prices. We shall refer to the n = 0 case as the privatized system. We will not focus on the privatized system since it is a polar allpayer system. case of the much more general The allpayer system embodies a multipayer system in which the government, the hospital, and the private insurer engage in trilateral negotiations to set a uniform qualitybased rate without a passthrough payment. More specifically, instead of the government setting its rates before everyone else in stage 1 the government, together with the hospital and private insurer, negotiates a rate after the hospital selects T trilateral Nash bargaining solution * and I. That is, the allpayer system, solves the program max# YQ(I, T)WV"()MZ(#, 0) such that Y, V, M E [0, 1] such that q + x + system is equivalent to the privatized system. 1. Note that when n=0 the allpayer This allpayer model is very similar to the allpayer systems found in Japan and in four American states in which a uniform fee is negotiated by all payers. However, what is unique about our allpayer model is that we consider a qualitybased uniform rate. That is, the fundamental differences between the singlepayer system and the allpayer system in our model are that the allpayer system sets the uniform price after observing both T and I, and, moreover, prohibits passthrough payments. We assume that it is beyond the ability of the government to monitor I and T in the the singlepayer system. We take this view of the singlepayer system since it is most reflective of the Canadian singlepayer system in which reimbursement rates are purely prospective and are rarely based on I and T (Merrill (1994), p. 251). The Overinvestment Problem where n, q, z, z optimal investment level. The socially efficient investment level TE and intensity level IE maximize the total expected social welfare U(I, T) c(I, T)X(I) which we assume to be concave in I and T. Before comparing the systems, we will define A(I,T = (1 n)U(I, T) c(I, T)X(I) + naX(I) + (p to be the total surplus that is to be bargained over by the hospital and the private insurer in the multipayer system. A is assumed to be concave in I and T. Next, it will be helpful to construct the following elasticities. technology The bargaining elasticity of T is defined as CTT = ETT(I,TIa,p) = TATT (2.1) Similarly, the bargaining crosselasticity of intensity I is defined as CIT = CIT(I IATI ,TIc,p) =  (2.2) see that cTT is an elasticity, observe that equation (1) can be rewritten as CTT d(ATr) Ar dT Thus, CTT gives the percent change in AT, the marginal bargaining value of tech nology T, over the percent change in T. Similarly, CIT is the percent change in the marginal bargaining value of technology, when the treatment intensity I is changed, over the percent change in I. comparison of elasticities'3. Throughout the paper we will maintain the following Assumption 1 will hold whenever treatment intensity I and technology T sufficiently strong operating cost complements. For most inpatient procedures it is Assumption 1 1)T operating cost of an additional unit of treatment intensity decreases as the medical equipment accompanying the procedure becomes more advanced. If I and operating cost substitutes, utility complements (UIT then Assumption 1 Next will hold if I and T display sufficient , we need the following additional assumption. Assumption U' This assumption is sufficient to ensure that the A 0 constraint is nonbinding in equilibrium. This will simplify the equilibrium analysis n the following comparison of the three health care systems. Proposition 1 When passthrough payments are allowed: In the singlepayer system equilibrium (Ts, Is, in technology (Ts the hospital will underinvest of treatment intensity (Is use passthrough paym ents (ps . Moreover, the government will choose not to = 0). The hospital will earn nonnegative profits (Is, Ts I). Under bargaining, the allpaye r system will provide the hospital with the incen tive to invest efficiently at T less of its negotiating power q. E and to supply the efficient intensity IE hospital will earn nonnegative , regard profits of (IE, TEO). In the multipayer em equilibrium, (To, o,,po), there exists a critical market share no ( TE) if n < regardless 0, 1) such that the hospital will overinvest in technology (To(n) o. The hospital will undersupply treatment intensity (lo(n) < I of its negotiating power q. TV/en sa the government will use a passthrough payment to pay for po percent of the hospital's technology, where =min t  TnAr, 4 A ,r .4 n. 'l (1'I.R < TE) and will provide less than the socially efficient level I Irl 18 These results are illustrated in Figure 1. Since the MAReffect has been removed by decoupling demand from technology, the hospital has no incentive to invest in tech nology (beyond some mandatory minimum threshold) under a singlepayer system. The reason for this is that the payment system is "too prospective" . To induce invest ment in technology, a complete passthrough (p=l) is not enough, the government would also have to reimburse a portion of operating costs retrospectively (this will be addressed in the next section). Similarly, the payment is also still too prospective with respect to the demandinducing treatment intensity. in the literature (Glazer and McGuire (1994), This is a common result Ellis and McGuire (1986,1990)). However, Proposition 1 reveals a more interesting consequence of prospective pay ments that warrants special attention. Among the three systems, overinvestment in technology is only induced by prospective payments under the multipayer system. The overinvestment results from externalities created by the uncoordinated dual reg ulation of the hospital. see this, first note that the hospital now has an incentive to invest in technology in order to increase its bargaining position with the private in surer. Next, since the investment is not hospitalspecific, ex post negotiations do not result in the holdup problem of Klein, Crawford, and Alchian (1977) and Williamson (1975,1977). Thus, the private insurer cannot expropriate the investment. Therefore, under the allpayer system, this investment is socially efficient since the public payer cannot externally influence the bargaining process. However, under the multipayer system, the public payer imposes an externality on the negotiation process. More precisely, the public payer engages in costshifting by employing an "inverse elasticity rule." That is, since technology is more inelastic than intensity (CIT > rTT), the public payer increases the passthrough p (with respect to the singlepayer Ps) in order to increase reimbursement of T and lowers a to decrease the reimbursement for I. As a result, the hospital now has the incentives to overinvest S v r^/n nrnlvn r v rnla .,n A +n f,.C r vmA n.nlr 4+ r + wnn lifl+Of Oi+ (p=l) (p*) m*m  * ALLPAYER MULTIPAYER SYSTEM SYSTEM SINGLEPAYER SYSTEM FIGURE 21 : EQUILIBRIUM INVESTMENT This result may well characterize the present health care market under prospective rate setting. It is widely agreed that prospective payments from Medicare as well as aggressive payments from HMOs induce too low of a treatment intensity, while technology costs continue to soar (Newhouse (1992), Thorpe (1992)). As poten tial evidence of this overinvestment problem in multipayer systems and the underin vestment problem in singlepayer systems, it is interesting to note that Atlanta has more Magnetic Resonance Imaging facilities (MRIs) than all of Cananda (Newsweek, July 25, 1994). At the beginning of Medicare's Prospective Payment System (PPS) in 1984, Atlanta had only one MRI facility. Now it has at least thirty (Eckholm and Pear (1993)). Overall, the U.S. has eight times more MRIs than does Canada on a per capital basis (Starr (1994)). Finally, if we interpret I as patient length of stay, it is interesting to note that the U has the lowest average length of stay among OECD nations, with an average of 7.1 days, compared to 12 to 24 for most other nations (Reinhardt (1992)). Finally, we note how the results of Proposition 1 stand in contrast to the results of Ma and McGuire (MM) (1993). ating costs are constant, that de MM assume that intensity is exogenous, that oper mand is induced by technology (an MAReffect), and that the private payer can set rates only before the hospital selects T. Under these assumptions, MM find that a multipayer system results in technological underin vestment. Moreover, MM's singlepayer system is efficient. These two results do not seem to support the general empirical evidence. In contrast, our characterizations of the multipayer and singlepayer systems seem to be more realistic. Moreover, in MM's model, the government will implement complete passthrough payments (p=l) in both the multipayer and singlepayer system. In our singlepayer model the government completely dismantles passthrough payments. our multipayer model supports the use of an interior passthrough In addition, p E (0,1) in *1I 5 r 5. payment will also diverge from Ma and McGuire's policy recommendations. This will be explained more fully in Section 6. Mixed Payment Systems In the preceding section all government reimbursements of operating costs were paid prospectively. That is , a fixed per patient rate was set in advance. Proposition 1, we saw that this led to equilibrium rates that were "too prospective" in the singlepayer system, leading to underinvestment, even when the costbased passthrough payments could be made. In this section we show that the singlepayer system can be efficient when the government is allowed to use mixed payments in which a portion of operating costs can be reimbursed retrospectively. In the spirit of Ellis and McGuire (1986), mixed payment plans are defined by the expanded payment plans (a,p,r) and (f, R), in which the public insurer pays an additional rC(I,T) per patient and the private insurer pays an additional RC(I,T) per patient. As the next Proposition demonstrates, these mixed reimbursement rates have very different consequences under each health care system. Proposition 2 Under mixed payments'4: The singlepayer system induces the hospital to invest efficiently in technology and to provide the socially efficient level of treatment intensity. The govern ment's payment is completely costbased: a 1, and r In the multipayer system equilibrium under mixed payments, (Ti II,pt), there exists nl c (0, 1) such that the hospital will select TE and IE for n > na, but will overinvest at the level Ti(n) > To(n) > TE and will undersupply intensity, II(n) = Io(T,)(n) when n Moreover, r > 0 and R =0. 22 9. The allpayer system admits an equilibrium that is efficient and uses no cost based payments (R = 0). This Proposition reveals an important disparity between the singlepayer system and the multipayer system when the government employs mixed payments. Mixed payments resolve the underinvestment problem that arises when prospective pay ments alone are employed in the singlepayer system (recall Proposition 1). Not only is it in the government's best interest to use partially costbased payments, but it is socially optimal. In contrast, the government's actions are no longer aligned with society's best interest under a multipayer system. Instead, the government will now overemploy r, the costbased element of the payment, in order to shift costs to the private insurer. In fact, r gives the public payer even more costshifting leverage than it had under the prospective payment plan. As a result, the overinvestment problem of Proposition 1 is exacerbated under mixed payments. The main result of this section is that mixed payments increase the degree of costshifting in the multipayer system. Glazer and McGuire (GM) (1994) find a similar result in a related multipayer model. In the GM model, fixed costs (tech nology) are exogenous so that passthrough payments are unnecessary. Furthermore, intensity does not induce demand. Moreover, actual costs are not contractible; only allocated costs are contractible. this setting, GM show that both payers use a mixed payment (in which the costbased rate is approximated by a costallocation formula), inducing the profitmaximizing hospital to undersupply intensity. Here, when technology is endogenous and actual costs are contractible, the hospital will overinvest in technology and still underprovide treatment intensity. This finding reveals that costshifting does not necessarily result from an inability to contract on actual costs. Costshifting can arise when actual costs are contractible if the private payer does not use a mixed payment in response to the government's mixed payment. 23 Our result on the efficiency of mixed payments in the singlepayer system is not new. Mixed payments have been advocated quite regularly for singlepayer systems (Ellis and McGuire (1986,1993), Goodall (1990), Ma (1994), Pope (1989)). However, in the next section we show that goldplating is possible. this traditional result may be overturned when Goldplating In general, may often include not just expenditures on technology, but the entire capital costs of constructing a new facility to house the technology (e.g., a new wing on the hospital for an openheart surgery center). So far we have consid ered T to be a productive asset in that it had patient value U(I,T). In the preceding analysis, it was implicitly assumed that the government monitors the hospital's in vestment T to insure that nonproductive hospital assets with no value to patients are not bought and reimbursed through the direct passthrough payment. However, in reality Medicare and state Medicaid agencies often do not have the resources to monitor the capital expenditures of hospitals. Indeed, state certificateofneed laws, mandated by the National Health Planning and Resources Development Act of 1974, required (1985)). states to establish agencies to regulate hospital investments (Simpson However, many states have since abandoned certificateofneed laws since they have failed to prevent excessive investment (Sloan (1988), Salkever and Bice (1976), Joskow (1981))'5 This problem is not restricted to the state level. in no way rewards prudent behavior,' says Care Financing Administration (HCFA), " 'We've set up a system that Gail Wilensky, former chief of the Health which runs Medicare. 'In fact, the more you spend, the more you get.' And it doesn't much matter what the money buys. re (Medicare) pays as willingly for a lobby waterfall as for an innercity emergency room"(Business Week, April 22, 1991). When a hospital's capital expenditures go unmonitored while being largely reim bursed with direct passthrough payments, the hospital may find it advantageous to invest in capital that has value to management and doctors but not to patients. For example, hospitals often build elaborate offices for doctors adjacent to the hospital. Other examples of goldplating include elaborate architectural facades, overly plush doctors' lounges, onsite athletic centers and childcare centers for the hospital staff, elaborate artwork for private rooms, and general managerial slack. Unfortunately, the patients do not receive any direct benefits from these managerial resources, even though they may be subsidized by Medicare. One way to resolve this moral hazard problem is to set up proper managerial incentives and ex post utilization reviews (as analyzed in Encinosa and Sappington (1995), for example). However, the next Proposition shows that goldplating can be deterred simply by restricting passthrough payments, without adversely distorting the technological investment incentives. Let us assume that a nonproductive hospital capital expenditure of G has value b(G) to management and zero value to patients. Moreover, we will suppose that a hospital which is vertically integrated with payers would never invest in G. That is, we assume G has negative net present value in isolation, i.e., '(0) 1 and 4" < 016 Note that if 4' = 0, the hospital will choose G=0. That is, the hospital will never waste resources. However , when > 0, then the hospital will choose a positive G under a multipayer system. This is referred to as abusing resources, following Blackmon (1992)17 Proposition 3 capital expenditures are not monitored by the public payer and if mixed payments are prohibited, then: 16We are assuming that operating costs are independent of G. An interesting problem that we  .  ., a a i t In the multipayer system, solves the equation C"(G)[1 Po  GArr]  "()'(G)  ATTP'(G) + ATT (2.4) where is the passthrough payment in the monitored case of equation Moreover, the hospital will underinvest in technology if passthrough payments are prohibited. There is no goldplating (G n t/i nglepayer and the allpayer systems. In general, it quite difficult to verify whether equation (4) has a positive solution. However , if 4 is quadratic, then we have the following existence result. Corollary 1 If 4(G) in the multipayer system (without mixed payments), the hospital abuse resources at the level G* 22 62ALIT > 0. Moreover, the government will scale back its passthrough payment to p* = bG* forcing the hospital to scale back technology from the To level. Goldplating arises only in the multipayer system since passthrough payments are instituted only in the multipayer system when mixed payments are prohibited Proposition 1). Note that in isolation , the return on asset G is always negative (since 4' < 0). However , once a positive passthrough payment is introduced, there is always an asset level G that earns a positive return (4'(G) 1+p > 0). If the public payer cannot monitor the hospital's investment in order to weed out unproductive assets , Corollary 1 indicates that it is then optimal for the government to scale back its passthrough payment. Note that the passthrough payment is never fully dismantled. in equilibrium. Thus, goldplating, or abuse, is never completely eradicated The reason for this is that the government's ability to shift costs is eroded without the use of the passthrough payment. In essence, the moral hazard  '(G*) if there exists a positive G* G merit to scaling back passthrough payments: the hospital will also scale back its overinvesting in technology. In Proposition 3, goldplating did not arise in the singlepayer system since pass through payments are not used in that system when mixed payments are prohibited (see Proposition 1). However, recall from Proposition 2 that the government will im plement a complete passthrough (p = 1) of all capital in the singlepayer system once a mixed payment is allowed. As a consequence,the singlepayer system is susceptible to goldplating under mixed payments. Corollary Under mixed payments, we have G > 0 and p, r E (0,1) in the single payer system, resulting in underinvestment in technology and underprovision of treat ment intensity. The results of Corollary 2 stand in contrast to the recent strand of literature which suggests that singlepayer systems are efficient under mixed payments (Ellis and McGuire (1986, 1993), Goodall (1990), Ma (1994), Pope (1989)). These studies do not address goldplating. The salient finding of Corollary 2 is that if goldplating is a problem as the empirical evidence indicates, then mixed payments cannot restore efficiency in the singlepayer system. back both its costbased rate and its passthrough payment. The government must tradeoff less technology and less treatment intensity for less goldplating. Together, Proposition Corollary reveal a singlepayer system will persistently curtail socially efficient technological investment and treatment intensity. Public Policy Imolications When the Prospective Payment System was first introduced in 1983, it is conceivable that Medicare failed to anticipate the incentive for hospital goldplating To deter goldplating, the government must scale 27 Congress due to intense hospital lobbying. It was not until 1991 that HCFA succeeded in persuading Congress to force Medicare to phaseout passthrough payments over a ten year period. Corollary provides a possible theoretical justification for this HCFA policy of prohibiting the costbased passthrough payment. Although many have thought that the HCFA's primary motivation for a phaseout of passthrough payments was to end a federally financed "medical arms race" , we have shown that the reform actually deters goldplating as well as eases the overinvestment externality of dual regulation that persists in the absence of an "arms race" Note that Corollary 1 reveals that it is not in HCFA's best interest to lobby for a full dismantlement of the passthrough payment. HCFA to scale back the passthrough payment, it i passthrough a small fraction of capital costs. Inde< to maintain some degree of passthrough payments. While it is indeed optimal for is always optimal for HCFA to ed, it is even socially beneficial While a complete prohibition on passthrough payments will completely deter all goldplating, it will nevertheless induce the hospital to underinvest in technology and underprovide treatment inten sity. Note that (1993), this policy recommendation does not go as far as Ma and McGuire who advocate a complete restoration of the passthrough payment (p=l) in order to ease the severity of the underinvestment. MM's Goldplating is not consider in model. If it were, their model might also recommend an interior passthrough p . While the passthrough payment reduction of Corollary 1 does not necessarily lead to efficiency in technology and intensity in the multipayer and singlepayer systems, it is interesting to note that the hospital would perform at the efficient levels if the public payer charged the hospital a fixed licensing fee (or entry fee). Moreover, the 1 1 1 1 1 (i Proposition A If the public payer is allowed to impose a fixed licensing fee L on the hospital for any purchase of technology, then: The singlepayer system induces the hospital to invest at the socially efficient technology level TE and to provide the efficient level of intensity IE under an equilibrium fixed license fee L = Y(I, TE) The multipayer system induces the hospital to invest at the socially efficient technology level TE and to provide the efficient level of intensity IE under an equilibrium fixed license fee L = A(IE, TEn) Moreover, the public payer will still use a positive passthrough payment with L. This Proposition is not surprising. As expected the public payer uses L to extract all of the surplus from the hospital and the private insurance sector. This full rent extraction leads to efficiency since L allows (a, p) to be adjusted to essentially sellout Medicare to the hospital and the private insurance sector. This sellout equivalently resets n to 0. Thus , as in the allpayer case (or privatized case), it is in the interest of the hospital to perform efficiently (Proposition However, Medicare benefits because it can then extract all of the surplus from the sellout via the license L. Note that efficiency is obtained without the private payer charging a licensing or entry fee. Interestingly, this licensing fee policy is in contrast to the twopart hospital tariff policies of GalOr (1994) and Ma and Burgess (1993). GalOr shows that overin vestment may be the result of hospital competition under stochastic demand in a singlepayer setting. Ma and Burgess demonstrate that suboptimal intensity may result from hospital competition. However, both papers reveal that efficiency can be restored if the single payer provides the hospitals with lump sum subsidies. contrast , our model advocates the use of licensing fees when there is no hospital  29 purchases. However, Proposition 4 may reveal why these certificateofneed laws have failed to contain costs. First, the efficiency results of Proposition 4 work because it is the public payer who collects the entry fee. This allows the public payer to adjust (a, p) in a way that does impose an externality on the hospital. In contrast, many states to control tl have set up independent regulatory he licensing of new capital for hospital health systems agencies (HSAs) als. The HSA licensing fees are set independently of Medicare and Medicaid's reimbursement decisions. According to Reinhardt (1992), HSAs are completely divorced from the reimbursement decision. Proposition 4 indicates that efficiency obtains only if Medicare has complete jurisdiction over setting entry fees and licensing fees. A second problem with the HSAs is that they have tended to set fees that are substantially below the optimal fee L recommended by Proposition 4. For example, in Florida the application fee to request permission to open an openheart surgery center is $ 22,000. However, the hospital can make this back in the first 15 minutes of the first operation they do (The Tampa Tribune, September 24, 1994). The fixed fees required by Proposition 4 are extremely large in that they must extract the entire surplus from the hospital and the private insurance sector. Such large licensing fees are obviously not feasible in reality. Instead, Proposition 1 offers a more viable alternative in order to obtain efficiency: the allpayer system. It is interesting to note that in 1977 the Carter Administration proposed a series of all payer revenue controls on hospitals. However, after three years of legislative battles, the initiative was defeated by intense hospital lobbying (Ginsburg (1988)). About the same time, allpayer systems were tried successfully in Maryland, Massachusetts, New Jersey, and New York. Medicare, Medicaid, and private insurers cooperated in setting a uniform rate (Thorpe (1993)). int;l tbh 1QRn'A Hospital costs were controlled somewhat 117b0~rrl~~ n ~ m t ~ NiA,r ~xifAn axr a Vrrnrn #b00 tllnota cvl .a+ otonrn :a since it can shift costs to the other payers. As we have shown, this costshifting under the multipayer system will encourage excessive technological investment. Although the uniform rate structures in the singlepayer system and allpayer system remove this costshifting externality of the multipayer system, only the all payer system induces the hospital to perform efficiently. The reason for this is that the allpayer uniform rate is qualitybased, linking the hospital's choice of I and T to its reimbursement. In fact, qualitybased reimbursement has recently become an issue in the public sector. For example, Oregon recently passed legislation to link the approval of capital projects to the hospital's patient outcomes (Alter and Holtzman (1992)). In addition, New York's Commissioner of Health , Mark Chassin, recently proposed linking hospital reimbursement to qualityofcare measures in the 1993 renegotiations of the state's prospective hospital reimbursement system (Darby (1993)). The key result of this paper is these new aggressive qualitybased reimbursement mechanisms will only work efficiently in the allpayer system in which costshifting and goldplating are both eradicated. Conclusion The results of our theoretical paper suggests some important directions for future empirical work. First, how has Medicare's phaseout of the passthrough pay ment affected hospital investment? Second, what is the degree of cost complements between I and T (clT < 0) and what is the degree of EIT? Lastly, how hospitalspecific and redeployable is hospital technology? If hospital capital is irreversible, then the allpayer system will result in underinvestment due to Williamson's (1985) holdup problem. In such a case, the multipayer system may be more efficient, with the overinvesting problem mitigating the holdup problem. CHAPTER 3 OPTIMAL HOSPITAL CAPITAL STRUCTURE Introduction The most hotly debated public policy issue during the Clinton Administration is the financing of health care. It is widely agreed that any health care reform must first address cost containment. In 1990, expenditures on health care in the U.S. exceeded 12 percent of GNP. Medical costs are projected to soar to 36 percent of GNP by 2020. The key to successful reform is to understand the source of this rise in health care cost. Unfortunately, in attempting to explain this explosion in medical costs, many economists have failed to understand that the financing of health care is subject to a fundamental paradox. The health insurance market (composed of a public sector: Medicare and Medicaid, and a private insurance sector) attempts to set reimbursement rates to cover the hospital's cost of capital and technology. However, any reimbursement plan affects the hospital's expected future earnings, turn, affects the hospital's cost of capital. which, in The circularity of this process suggests that both the hospital and the health insurance market should take into account the simultaneous determination of the cost of capital and the prices of health care services. This paper lays a foundation for the equilibrium determination of hospital capital costs and medical insurance reimbursement. This chapter is organized as follows. In Section 2, we explain how the capital market prices the hospital's financial securities. In Section 3, we examine the strategic role of hospital debt. All proofs are relegated to the Appendix. The Model begin we consider a stochastic version of Ma and McGuire's (1993) model of the demand side of the health care market. For simplicity, we assume that consumers in the health care market are fully insured1 Thus market demand is assumed to be price inelastic with respect to the price of medical services. However, market demand depends crucially on the level of advanced technology and stateof theart care offered by the hospital. Letting T be the hospital's monetary level of investment in technology, we define X(T ,z) to be the number of patients who demand admission to the hospital in the realized state of nature of acute medical services is highly stochastic. This vol z. The demand for most types latility will be modeled by the demand shock levels of z E [0,1] z correspond generated to greater demand by the continuous distribution F( so that X(T ,z) is increasing in Higher z for all We assume that it is illegal for the hospital to ration services. Thus , capacity (e.g., number of beds) is fixed at a sufficiently large enough level so that health care services are not rationed in equilibrium, even when demand is most pronounced (i.e., when z=1)2 Since there is no rationing, X(T,z) can also be considered as the realized number of discharges from the hospital. Demand for hospital services X(T is assumed to be increasing and concave in technology investment T Finally, n will be the fraction of patients that are insured by the public payer (Medicare, Medicaid); the other (1n) of the market is insured by the private payer3. For simplicity, we do not allow patients to be covered by both insurers. On the supply side, the hospital has a constant operating cost of c per discharge. The public payer pays c per discharge plus a margin a per discharge to help pay for 'We wish to show that overinvestment can occur under universal coverage. For a discussion of the problems in supplying medical insurance, see Diamond (1992) and Lewis and Sappington (1994). 2We are primarily concerned with the optimal T when there is a threat of bankruptcy when z is T. In addition, the public payer can make a passthrough payment p(T + i) that is independent of the number of discharges, where pE [0, 1] and where i is the riskfree interest rate. This payment plan represents the Prospective Payment System (PPS) administered currently by Medicare and by some state Medicaid programs (ProPAC 1993)4 The private payer will pay c plus a margin (f per discharge. 6 will be determined by negotiations between the private payer and the hospital. Historically, passthrough payments have never been instituted by private payers, and so will not be considered here. Our model of the health care system focuses on the interactions among four ac tors: the public payer, the hospital, the capital market, and the private payer. timing of the game among these players is as follows. In stage 1, the public payer (Medicare) institutes the margin a and the passthrough payment parameter p. stage 2, the hospital chooses the level of investment T, and a mix of equity and debt to externally finance the investment T by issuing new shares and bonds. At stage 3, the capital market determines the market value of the hospital's securities. Finally, in stage 4, the private payer's margin p arises out of negotiations between the hospi tal and the private payer as the Nash bargaining solution. Then the demand shock z is realized, medical services are delivered as demanded, and payments are made, with bankruptcy being declared if necessary. The exact details of each stage will be discussed next. Stage 1: The Government's Payment Plan. The public payer is the first institution that sets prices, establishing the margin a and the passthrough payment parameter p. Medicare is presumed unable to update its payment plan after the hospital makes its investment, perhaps due to a lack of managerial resources. Thus, while anticipating the results of the next three stages of the game, the public payer selects a reimbursement plan (a, p) in order to maximize the expected net social benefit to the segment of the market it serves: M = M(a, p) = n[U(T) (c + a)X(T)] pT(1 + i), where X(T) = X(T , z)dF( where U(T) is the gross social benefit when the hospital provides technology level T. Note that i is the riskfree interest rate. Stage 2: The Hospital's Financial Strategy The hospital is assumed to maximize its profits. can easily be constructed. A nonprofit analog of our model We choose to focus on the forprofit case since the number of forprofit hospital beds increased by percent between 1976 1986, while the total number of hospital beds declined by 10 percent during the same period (American Hospital Association (1987)). The hospital is assumed to have no retained earnings and no previous debt obli gations. It is owned by the shareholders of a previous issue of voting equity (common stock). These shareholders will be referred to as the "old" stockholders. Next, the hospital invests dollars in technology and finances its investment from external sources. When new shares are issued to new shareholders, s'C1 will represent the fraction of the hospital's total number of shares that these new shares comprise. It is also possible for the hospital s<0). to repurchase some of its existing equity (i.e., The hospital can also issue debt with a face value of D. The hospital chooses both the magnitude of the technology outlay T (s,D) = T(a, p) and the capital structure = (s(a, p), D(a, p)) to finance T. In doing so, the hospital anticipates the out come of the next two stages of the game. The presumed autonomy for the hospital 35 insurers typically do not have the expertise to specify technology and security design for the hospital in detail. The operating income of the hospital is R(a, /, p, T, z) = [na + (1 n)] X(T, z) + pT(1 + i). As the hospital takes on debt, there may be a critical value of the demand shock z*, below which demand is too meager to enable the hospital to pay all of its debt. This critical value is defined by  min{z R(a, 3, p, T When the demand for hospital services is sufficiently low that (3.1) z < z* limited liability applies, the hospital declares bankruptcy, and the bondholders become the residual claimant. When demand is sufficiently high that the hospital remains solvent, and both the old and new shareholders remain the residual claimants. Note that F(z*) is the probability of bankruptcy. Bankruptcy imposes costs on the bondholder due to legal fees, reorganization, and disruption of services. Moreover, a financially distressed hospital may cut back on the quality of care, increasing the risk of malpractice. Following Brander and Lewis (1988), we assume that bankruptcy costs are proportional to the size of the shortfall in the hospital's earnings from its debt obligation. That is, realized bankruptcy costs are b[D(a, p) R(a, /3, p, T, z)] for some positive exogenous cost b per unit of shortfall when z< z* Since bondholders are also protected by limited liability, we assume that the hospital is liquidated whenever b[D(a, p) R(a, /, p, T, z)] > Let z** be the critical value below which the hospital is liquidated. , the hospital is not liquidated, but is reorganized R(a, /, p, T, z). When under the ownership of the bondholders who receive R(a, /, p, T , z) b[D(a, p) R(a, p, T, z)]. Thus, expected Given the hospital's financial securities is risky debt obligation D, the total expected return on Il(a, /, p,T) = R(a, /, p, T, z)dF(z) L(a, /, p, T). These expected profits are the combined expected returns to shareholders (old and new) and bondholders, divided between them according to their respective claims. However, hospital management will select expected profit to the old shareholders s: and D in order to maximize the =Y(T ,s,D,E,B) = (1 s) [R(a, /, p, T, z) D(a, p)]dF(z). (3.2) Stage 3: Next, The Capital Market. the capital market establishes the market value E of the hospital's new equity and the market value B of its debt with face value D. We assume that the capital market is riskneutral, competitive, and that investors correctly anticipate the outcome of the private insurer's negotiations with the hospital over the reimbursement margin p. The hospital's securities are priced fairly so that both the new shareholders and the bondholders earn an expected return equal to i, the riskfree interest rate. That is, in equilibrium, the capital market sets the following valuations: (1 + i)E [R(a,, p, T ,z) D(a,p)]dF(z), (3.3) (1+i)B D(a,p)(1 F(z)) + R(a, /3, p, T, z)dF(z) L(a, f, p, T)(3.4) with E + B While the market only requires E + B (3.5) we assume the private insurer can prohibit the hospital from accumulating discretionary cash flows. Thus, equation (5) is assumed to hold. In equation (3), the value of the equity is equal to the fraction of the expected operating income of the hospital that goes to the new stockholders, net of debt payments when the hospital is solvent. The first term on the right side of equation (4) is the expected return to bondholders when demand is high enough to ensure that they are paid D n full. The last two terms represent the expected operating income of the hospital when it becomes financially distressed due to low demand, net of bankruptcy costs. Combining equations (3),(4), and (5), we obtain the capital market equilibrium condition: (1 + i)T(a,p) = D(a,p)(1 F( R(a,, p, T,z)dF(z) L(a,, p, T) + +sJ The hospital's equation (6). R(a, p,T,z) D(a,p)]dF(z)(3.6) optimal capital structure (s, D) will always satisfy the capital market Solving for s in equation (6), the hospital profits in equation (2) can now be expressed as 1 Jo R(a, f, p,T , z)dF(z)  T(1 + i) L(a, p,T). Stage 4: Negotiations with the Private Insurance Sector. Next, we model the private insurer as a managed care facility, such as an HMO, engaging in utilization review. That is, we allow the private insurer to negotiate the margin p with the hospital after observing the hospital's investment T, capital struc ture (s,D), and we will derive /3 party the market valuation (E, B). To capture this aspect of negotiation, as the generalized Nash bargaining solution. 's utility function. First, we specify each We assume that the private insurance sector is competitive *>> t to the patients it represents:6 V(3) = (1 n)[U(T) (c + 3)X(T)]. When the hospital management goes to the bargaining table, it will still represent only the old stockholders, seeking to maximize Y from equation (2)7 . If the hospital backs away from the bargaining table, its only income will be from serving the patients insured by the government. That , the hospital's threat point is endogenously determined by the public insurer's reimbursement plan. Note that if the hospital fails to negotiate (i.e., = 0), it will prefer to serve the publicly insured patients rather than to close the hospital. serves only the public patients, However, if the hospital then the capital market will devalue the hospital's securities from (E B) to (Eo, Bo) (where the 0 subscript refers to the f/ = 0 case): (1 + i)Eo 4 (naX(T,z) + pT(1 + i) D)dF(z (1 + i)Bo (naX(T ,z) + pT(1 + i))dF( where /0 (naX(T, z) + pT(l +i))dF(z)+ &J .[D a"  (naX(T, z) + pT(1 + i))]dF(z = min{z naX(T,z) + pT(1 + i) = min{z Although the hospital naX(T,z) +pT(1 +i)}. will prefer to serve the public sector when P bondholders and the new shareholders may instead want the hospital to liquidate T and pay back as much of B and E as possible. We will assume that the discounted resale value of T is 7T , where 7 E {0,1}. When 7 = 1 the technology is completely 6For simplicity, in V and M, we assume that patient welfare does not decline when bankruptcy occurs. However , welfare is affected indirectly by the risk of bankruptcy via 3. In the stage 4 subgame, there is no conflict of interests between the old stockholders, the new  (naX(T,z) + pT(1 + i))] > + redeployable. In contrast, when 0, the capital asset T is hospitalspecific. most cases medical technology is not hospital specific. However, the accompanying infrastructure (i.e., a new wing of the hospital) may be irreversibly sunk. Moreover, 7 may include the litigation cost of suing the hospital to liquidate T. In particular, since Eo < E and Bo < B, the bondholders and new shareholders will always prefer suing the hospital to liquidate T when 7 = 1 (recall that T = E + B from equation (5)). However, the investors cannot completely recover resale value or when litigation is too costly (7 = 0). keep their downgraded securities E0 and Bo. when the asset T has no In this case the investors will threat point for the bargaining game is Consequently, the old shareholders' r(7)= (17)(1 s) Next, the private insurer has no purchasing power with outside hospitals. Thus, we will normalize the private insurer's threat point to 0. So, with a (r(7), 0) threat point, the generalized Nash bargaining solution Jf* maximizes (Y r(7))Vl'q while guaranteeing V, the hospital.  r(7) where q E (0,1) is the relative bargaining power of Debt Strategies Suppose that there are no costs to bankruptcy (b reversible (7 = 1). = 0) and that technology is In such a setting the hospital always chooses an allequity capital structure. However, once we introduce bankruptcy costs (b > 0), the hospital's optimal capital structure will now involve debt. When bankruptcy costly (b a positive level of > 0) debt D* and technology is reversible (7 = > 0, such that the probability of (naX(T, z) + pT(1 + i) D)dF(z) for Proposition 5 the hospital issues further. The Nash bargaining solution P* satisfies where X(T) = X(T, z)dF(z). increases in debt D* This debt strategy warrants special emphasis in our taxfree setting. Though we have focused on forprofit hospitals, this debt strategy can be shown to hold for nonprofit hospitals as well. In practice, since there is no applicable corporate income tax for notforprofit hospitals, we would expect nonprofit hospitals to maintain an allequity structure in order to avoid the risk of bankruptcy8 . Yet both nonprofit and forprofit hospitals are highly leveraged relative to most other industries. Proposition 6 offers an interesting explanation of this phenomenon. First, since the private insurer enters price negotiations with the hospital after having observed the hospital's capital structure and investment level, the private insurer can behave opportunistically in the negotiations, as we have seen previously. pital precommits to a highly levered position. To counter this opportunism, the hos This increases the risk of bankruptcy. Since bankruptcy is now costly, the hospital now demands a higher margin Pf. since both parties have equal bargaining power, the private insurer must concede to some increase in /. According to Proposition 6, the hospital's gain from this increase in the price per patient discharge outweighs its inflated risk of bankruptcy9 Conclusion Though bankruptcy is rare in most regulated industries (e.g., public utilities), bankruptcy is not uncommon in the health care sector10 . The reason is that the 8See Kraus and Litzenberger (1973), Scott (1976), and Flath and Knoeber (1981) for a discussion of the tax advantage of debt. 9Note that this result is very general in that we did not include a direct cost of bankruptcy in Moreover, the rate *P X(T) ~(lt b)V, X(T) health care market is much more stochastic than other regulated markets. As a result, future demand is difficult to forecast. Our model indicates that hospitals are willing to use debt in the shadow of this highly volatile demand in order to counter the opportunism of the private insurance market. CHAPTER 4 PRICING UNDER EXCLUSIVE DEALING Introduction Resale price maintenance (RPM) occurs when a manufacturer directly or in directly limits the price at which its product can be sold by an independent retailer. Minimum RPM occurs when the manufacturer sets a price floor by enforcing only a minimum resale price, allowing the retailer to charge a higher price. RPM occurs when the manufacturer sets a price ceiling. Maximum Since the repeal of state Fair State Trade laws in 1975, all forms of RPM have been 1S. However, many economists question this current pe se illegal in the United r se illegality standard, arguing that some forms of RPM may generate efficiencies that benefit consumers. This debate has resulted in a large economic literature that analyzes the motives underlying RPM. Yet, this body of theory has failed to explain two fundamental ob servations from the RPM litigation Trade laws. cases that emerged with the repeal of state Fair First, from the Federal Trade Commission Report by Ippolito [1988] which examined all RPM litigation from 1976 to 1982, it is evident that both mini mum RPM and maximum RPM are sometimes observed in the same industry across different markets1. Second, only exclusive dealerships were observed in the industries where both minimum RPM and maximum RPM were imposed. In contrast, common retailers were employed in almost all of the industries in which only maximum RPM appeared (i.e., rival manufacturers shared the same retailer2). This paper presents a model of manufacturer competition that explains these two findings concerning the connections between RPM and industry structure. 1For example, in 18 cases involving gasoline retailing, half involved allegations of price floors while the remainder involved rice ceilings. 43 In most manufacturing scenarios, retailers are often better informed about market demand than manufacturers. Furthermore, a moral hazard problem is often present because demand is influenced by the (unobservable) marketing intensity and pro motional efforts of the retailer. We construct a model characterized by asymmetric information and moral hazard and develop predictions consistent with the empiri cal findings of Ippolito. More precisely, we find that if the manufacturers compete through the same shared retailer, price ceilings emerge systematically for every real ization of demand. In contrast , if the manufacturers instead compete headtohead for the exclusive services of the retailer, price floors are optimal when demand is low (i.e., in weak markets) while price ceilings emerge when demand is high (i.e., in strong markets). Thus, the manufacturer's use of RPM depends critically on the industrial structure in a manner that is consistent with the empirical findings. There are other interesting differences that emerge in the incentive contracts under these two modes of manufacturer competition. First, when manufacturers compete through the same retailer, the resale price and sales level that emerge in equilib rium will always differ as the level of the market demand differs. In contrast, when manufacturers compete headtohead for the exclusive services of the retailer, a fixed, uniform retail price may emerge. In particular, the winning manufacturer institutes a rigid resale price that does not vary with demand in a range of intermediate demand levels. Thus, in a dynamic setting, an intertemporal price "stickiness" may emerge which is due entirely to healthy manufacturer competition. This finding stands in contrast to the traditional view that price rigidity is the result of tacit manufacturer collusion (Maskin and Tirole (1988) and Eaton and Engers (1990)3). Rigid prices emerge in our model because headtohead competition for an exclu sive dealership creates countervailing marketing incentives for the retailer. The usual i,n.+, inC~oF*t~o*t9nr *bn r or ra t ssrlomanr r trrlt~ .irtnror)a+ vm Ac 2nA 2Aaa +~r ffXtl'f b n2nlrtl that high sales are due to the retailer's diligent efforts, not exogenous demand. This incentive is countered by the following effect of competition. As market size increases, the total value of the retail outlet increases, leading to higher manufacturer bids for the retailer's services. The prospect of higher bids can encourage the retailer to ex aggerate the size of the market. These countervailing incentives give rise to partial pooling in the equilibrium contracts. For weak markets, the incentive to exaggerate the market size prevails, and, to mitigate this incentive, price floors are imposed. For strong markets, the incentive to understate demand dominates, and, to help control this incentive, price ceilings are instituted. In the intermediate market range, these two incentives begin to conflict, as do the price ceilings and floors. The result of this conflict is to enforce the same uniform resale price regardless of demand, provided de mand is of intermediate size. Consequently, in a dynamic setting, price adjustments will be more sluggish in the interim period between booms and busts than during booms or busts. This prediction differs from Maskin and Tirole's (1988) finding that price adjustments should occur least often during booms. The traditional explanation for RPM under manufacturer competition is that manufacturers use RPM to sustain collusion (Telser (1960) and Posner (1977)). contrast, we find that RPM is the result of intense manufacturer competition for a privately informed retailer. Although RPM under asymmetric information has been investigated in a setting with downstream retailer competition (e.g., Katz [1989] and Rey and Tirole [1986]4), little attention has been afforded to RPM resulting from upstream manufacturer competition. An exception is Perry and Besanko [1991] who examine RPM under manufacturer competition for exclusive dealerships. They ab stract from incentive conflicts and examine a restricted class of franchise contracts with RPM. They find that if manufacturers are restricted to use only fixed fees and/or linear wholesale prices with RPM, one equilibrium with maximum RPM arises, as does a second Paretodominating equilibrium with minimum RPM. Thus, their model predicts that minimum RPM will appear systematically under competition for ex clusive dealers. In contrast , our exclusive dealing game exhibits an undominated equilibrium that supports both minimum and maximum RPM in different markets at the same time This is consistent with the RPM pattern that Ippolito finds under exclusive dealing. Moreover , we do not restrict franchise contracts to any particular form. According to Ippolito's second empirical finding, the systematic use of maximum RPM is observed only in industries where retailers are shared by rival manufacturers. We develop a common agency retailing model in the spirit of Stole (1992), and Whinston (1992), Bernheim and Martimort (1992,1993) that supports this second finding on RPM. Although these authors study the choice between exclusive dealing and common retailing under manufacturer competition in detail6 , they do not examine the important connection between RPM and the industry structure. For example, Bernheim and Whinston (1992) allow resale price to be an unobservable decision of the retailer, and analyze a moral hazard common agency problem. In contrast, we assume resale prices are observable, and examine an adverse selection common agency model in the spirit of Stole (1992) and Martimort (1992,1993). Their models of manufacturer competition consider retailers with private information about the costs of supplying the final good. contrast The demand functions are common knowledge. In , we allow the retailer to possess private information about market demand. SThe singlemanufacturer analog of our model is developed by Blair and Lewis (1994). They find that both forms of RPM may emerge in a monopolized industry across different markets, but only when the standard SpenceMirrlees sorting condition fails to hold. This sorting condition ensures that the retailer is compensated less when demand is high than when it is low for a given increase in the sales level or resale price. In our model we show that both forms of RPM may emerge even when this sorting condition is maintained. It is interesting to note that Romano [1994] has shown that both forms of RPM may emerge under complete information, depending on an expanded elasticity nn WOll 2ia tb. rohtilia r mieQt mrnuil nn uunnhloawl,,vshla.i eltiaiann thnt mnrl;,;nn ;F thP m~nllflP)llrPP 46 As a result, our common agency model is multidimensional in the sense that both price and quantity must now be controlled by the manufacturer's contract since demand is not known ex ante by the manufacturers. This chapter is organized model. as follows. In Section 2 we develop the common agency We demonstrate that the equilibrium contracts result in maximum RPM in every market when manufacturers compete through a common retailer. rium contracts are nonlinear: The equilib simple twopart wholesale price contracts with RPM cannot be sustained in equilibrium. In Section 3, we contrast the distortions induced in a common agency with those induced by a multiproduct monopolist. Section 4 delineates conditions under which both forms of RPM may emerge when manufac turers compete headtohead for the exclusive services of the retailer. In addition, rigid retail prices are shown to emerge. Finally, Section 5 offers some concluding comments. The proofs of all the key findings are provided in the Appendix. 4.2 Common Agency In this section we will examine the case of manufacturers competing through the same shared retailer 7. Consider two vertically differentiated duopolists, M1 and M2, who wish to sell their products in the same market. in this market can be parameterized by a scalar 0 E 0 The demand for both goods to ~1 Higher realizations of 9 correspond to higher market demand in the sense described below. While 0 is unknown to the manufacturers, it is common knowledge that 0 has distribution F(O). In contrast, a potential retailer knows the demand realization 0 from the outset. The 7We are modelling a common agency with unknown demand. Recently, Stole [1992], [1992,1994], Ivaldi and Martimort [1993], Mezzetti [1993], and GalOr [1991a] have exar Martimort mined com mon agencies with unknown costs. Our common agency model is multidimensional in the sense that both price and quantity must now be controlled by the manufacturer's contract since demand is not known ex ante by the manufacturers. At the opposite extreme, Bernheim and Whinston  ~ a S era I 1 I iirl 11 U 47 manufacturers are assumed to offer take it or leave it contracts simultaneously to the retailer. On the demand side, we assume that the retailer cannot price discriminate. retailer sets a single (observable) linear price for customers. As a matter of exposition, we will first assume that the retailer's sales levels are observable. show later that the manufacturers' optimal contracts would not change if sales could not be monitored, provided resale prices are observable. The demand for product i is given by retailer's Ii p p e eL ) i ,2 Demand for product i increases with productspecific marketing intensity or promotional effort ei. Moreover, due to positive promotional spillover effects, demand for product i may increase in ei. For a large class of demand functions and disutility functions e(ei, e2), there will be a unique disutility minimizing effort pair (el, e2) which will ensure that (zl, zx), and no more , is sold at the price vector (pi, p2) in a market of size 08 . Thus, instead of specifying the demand functions directly, it will be more convenient to work with the welldefined composite disutility function e(xl, x2, pi,p2, ), which is the minimum cost or disutility incurred by the agent in insuring that (xl, x2), and no more, is sold at the price vector (pl, P2) in a market of size 0. As 0 increases, the level of effort needed to sell (x, x2) at price (p1,P2) declines. The retailer's effort may include marketing intensity, advertising, or such customer services as free delivery and installation, free repair or consultation services, and product demonstration. This type of effort is not readily observed by the manufacturers, so a moral hazard problem arises. That is, the manufacturers cannot verify directly whether low demand occurs because the retailer has supplied little effort or because demand is truly sluggish. On the supply side, manufacturer Mr produces zi at constant marginal cost mi, and imposes a tax Ai(xi, pi) on the retailer. Ax(z;, p9) is the amount the retailer must pay to manufacturer Mi when he chooses to sell zi units of Mi's product at price pi. Both pricing and sales decisions are delegated to the agent. Although price is not directly dictated by the manufacturer, she can indirectly control the price through the tax since prices are observable. Thus, manufacturer Mi seeks to solve the following program in the common agency duopoly game:" [CA] maximizeA,(,.)  miXi(0)} f(0)d0, where the multioutlay (zx, x2, p,p2) is chosen by the retailer to solve max,, z2,P2 {xipi + xZ2p2 e(x1,X2,pi,p2,) A1(ac,pi) A22p2)  The timing in the game is as follows. Next First, the agent learns the realization of , the manufacturers simultaneously offer the agent their tax schedules. agent then accepts either both, one, or none of the tax schedules. If the retailer rejects both contracts, he earns his reservation wage, which is normalized to zero. In this section, we will derive a perfect Bayesian equilibrium of this two stage game. Given the complexity of the game, a characterization of the equilibrium is provided for the case where the retailer's disutility of effort is quadratic, and takes the general form: e(a1, z2,p2, 2,0) 2 +h2x2p2 CZl 2 x2 + p + p+ hlxlxp +  2 2 2  cplpz 0(zl + z2 + pl + p). (4.1) Notice that the form in (1) admits an approximation of many disutility functions up to a secondorder Taylor expansion. and large enough that the retailer's dis We will assume that a; and bi are positive utility of effort must increase as the price or the required sales level increases, i.e., ex,, em, > 0 fori=l >0is a necessary condition for the retailer's program in [CA] to be concave in equilibrium. 9WpJ n n t I nllnw mnif tnrr un A'd, t fln ntr nrn tQ ft that nra, rntinevont wiTnnn di nantaa r*k,,, Also, aibi (1 hi)2 In addition, we will require that hi 1 for i=1,2. This natural assumption ensures that as the resale price increases, each unit of sales becomes more profitable for the manufacturer under complete information. will be substitutes if c Finally, note that the two product lines < 0 and complements if c When the manufacturers have complete information on product demand, equilibrium contract is extremely simple, consisting solely of a fixed franchise fee. This fee extracts all the rents from the retailer. But, more importantly, RPM does not occur under complete information. However , if the retailer possesses private information on product demand, the equilibrium franchise contracts become more complex and involve RPM, as the next Theorem illustrates10 . First we will make use of the following assumption throughout the paper. Assumption 3 0 has an increasing affine hazard rate. Due to the highly nonlinear nature of e, it is extremely difficult to develop an algorithm to compute the equilibria when the hazard rate of the distribution is also nonlinear. Under Assumption 1 we have the following existence and characterization result. Theorem 1 There exists a purestrategy equilibrium for the common agency game [CA]. In th equilibrium, franchise taxes take the form Ai(zi, pi) = 7i + cii + x] + SiPi + pi + OTizipi 2 p (4.2) for every (zi, pi) E31. that it always induces This the retailer to equilibrium schedule in (2) is fully separating in select an outlay (x, pi) that differs strictly with 1oDue to the moral hazard problem, the firstbest contract cannot be implemented as in the asymmetric demand information models of Lewis and Sappington [1988], [1992]. 1An increasing hazard rate ,' helps to ensure that the tax will be fully separating. The hazard rate is affine if 0 is generated by a Beta(1,1) density on 6 with distribution 1 4k4 for the realized market fee (7), Furthermore, the equilibrium taxes entail a fixed franchise a nonlinear wholesale price (a, /), a nonlinear tax on the retail price (6, 7), and a royalty (a) on gross revenue. Equilibrium royalty rates are ai i=1.2. There does not exist any simpler combination of a twopart wholesale price and/or royalty on revenue in equilibrium. The retailer's equilibrium profits increase with demand, When the products are substitutes (c the retailer earns rents even in the weakest market 0. Under complements, the retailer earns only his reservation wage in the weakest market _. Note the resale price is never directly imposed on the retailer, but it controlled indirectly. Resale price maintenance appears as a levy on resale price and a royalty on revenue. To understand the equilibrium pattern of rents, first consider the case where the two products are substitutes. In this case , the retailer naturally has an incentive to serve one manufacturer exclusively. Thus , the retailer has to be afforded extra partic ipation rents just to agree to serve both manufacturers. So, to induce participation, the retailer is allowed to retain a premium rent even in the weakest market 0. complementary products, Under the agent does not have to be forced to serve as a common agent, and so the retailer can be forced to zero rents in the weakest market13 To understand the nature of the royalties on revenue (ai), recall that h, is the degree of substitution between sales Xi and resale price pi in the promotional cost function or marketing function Since ai the royalty on product i's revenue increases at the same rate that the degree of promotional substitution between the resale price and the quantity of good i decreases. In particular, if resale price and 12Although taxes on the final resale price seem rare in practice, they have been advocated in the regulation literature. Laffont and Tirole [1990] and Leung [1994] have proposed using a tax on price to regulate a natural monopoly. Even though Ai is concave, it is still an open question of whether thars ovicto an nhnivhl nt aniilhrinrm in whih hn th rptanilpr nffor a mnnn riif affine fn ntreta onf thetP [ quantity are promotional cost complements of degree hi = 1, then no royalty is imposed on revenue. No revenue from the agent's ,te that this equilibrium royalty on revenue extracts all the sale of product i less the promotional cost term hixipi of e. Consequently, the fixed franchise fee, the nonlinear wholesale price, and the nonlinear surcharge on resale price must act as a combined subsidy to the retailer to recoup the remainder of his promotional costs and to provide any necessary rents. Next, we examine the more general case in which the manufacturers cannot ob serve the retailer's final level of sales. Below, Ai(xa,,pi), i=1,2, will refer to an equi librium of the common agency game [CA] in Theorem 1, where sales were monitored. Corollary 3 Suppose the manufacturers cannot observe the retailer's level of sales, so the retailer can sell any qi less than the ordered inventory xi, i=1,2. Then Ai(xi, pi), induce retailer to exact quantity that he orders, = xi. Moreover, the equilibrium franch taxes Ai(xi, pi), 1,2, indirectly impose price ceilings and quantity rationing on the retailer in every market 0. That is, Ai(xi,pi), i=l persists as an equilibrium franchise tax for the more general duopoly game where manufacturers do not monitor the retailer's sales lev In addition , Ai(xi, pi) does not give the retailer the incentive to destroy or to store excess goods. In fact the franchise taxes induce the retailer to choose resale prices and sales levels that are lower than what he would choose if he were the full residual claimant14 Thus, as a result of manufacturers competing through a shared retailer, the franchise tax indirectly imposes maximum, not minimum, RPM for ev ery realization of demand. This is in accord with Ippolito's examination of RPM litigation cases. In almost all the cases from 1976 to 1982 involving allegations of only maximum RPM, the retailer was shared by rival manufacturers. "The retailer would be the full residual claimant is if each manufacturer Mi offered a pooling The MonoDolv Benchmark As a benchmark, we now examine a monopolist who markets both xl and x2 through a single retailer when there is no threat of entry by a second manufacturer. To facilitate comparison with the above duopoly case, we make the following definitions. Let p?(0) and xD(0), ,2, be equilibrium resale prices and sales levels chosen by the retailer in the common agency (duopoly) game pM(0) and be the optimal resale prices and sales levels chosen by the retailer serving a multiproduct monopolist. Define pF(0) and zx(0), i=1,2, to be the corresponding (firstbest) levels for a monopolist who has perfect knowledge of demand. Similarly, define the equilibrium disutility of effort levels eD eF accordingly. We can now examine the efficiency of duopoly competition through a common retailer in the game [CA]. Proposition 6 Total manufacturer profits are higher under monopoly than under the duopoly of game [CA]. Furthermore: For substitute products (c pt'(0), <0), with equality xF(e) 4D(0) xy(0) and pf(0) market 0. pf (0) The price ceiling and quantity rationing are severe under duopoly. The retailer retains more rents under duopoly than under monopoly and works more efficiently: eF(0) eD(0) eM(0), with equality only for the largest market 0. For complementary products (c >0), 4(0() xM(e) F((0) and pD(0) p(0) ceiling pm(0), with equality only for the largest market 0. quantity rationing are more severe under duopoly. The price retailer secures less rents under duopoly than under monopoly and works less efficiently: eF(0) > eM(0) eD(0), with equality only for the largest market 0. Sy(0), only for the largest 53 (see Figure 3). Although a duopoly provides higherpowered incentives to the retailer than a monopoly would under substitutes, the total manufacturer profits are still higher for the multiproduct monopoly than for the duopoly. recall that for substitutes, only one manufacturer. TI To understand why, the retailer has a natural incentive to exclusively serve ius, he must be provided with extra rents as an incentive to serve as a common retailer for both manufacturers. In contrast, under a monopoly the retailer does not have this outside opportunity, so the monopolist does not have to afford the retailer these extra duopolistic participation rents. Price ceilings persist in the duopoly common agency game [CA] when products are complements. Furthermore, competition induces lower price ceilings and more severe rationing. Just as a monopolist will, the duopolists distort price and quantity downward as the market gets weaker in order to limit the retailer's incentives to understate demand. , because the products are complements, if one duopolist decreases her resale price and level of sales, it becomes advantageous for the rival to also lower her price and sales level. As a result, the manufacturer's tradeoff between efficiency and rent extraction becomes less severe as the rival reduces her price and sales level in order to extract rents. In essence, each manufacturer imposes a rent extracting externality upon the other when they share the same retailer, resulting in a double extraction of rents. Consequently, the retailer prefers a monopoly when the two product lines are complements. When the goods are substitutes , if one duopolist reduces her price and sales level, the rival will find it beneficial to increase both sales and price. Thus duopolist's tradeoff between efficiency and rent reduction becomes more severe as the rival reduces her price and level of sales in order to extract rents. Because of this externality under substitutes, the duopolist extracts less rents than the monopolist. As a result, incentives under a duopoly are higherpowered than the monopolist's 54 Finally, the underprovision of effort is worse under duopoly when the products are complements. Since a duopoly results in a lower level of sales at a lower resale price when compared with a multiproduct monopoly under complements, the level of effort required in a duopoly will be lower, and, level. hence The exact opposite occurs for substitute product 1 further from the first best ines; duopoly competition will induce the retailer to provide a higher, more efficient level of marketing intensity. Competition for an Exclusive Dealer In the last section we saw that when products are substitutes the retailer had to be afforded extra rents just to agree to serve both manufacturers. Eventually, these participation rents can become so costly for a manufacturer that her expected profits are higher if she instead hires the retailer as an exclusive dealer. To consider this possibility, we now analyze the case where, instead of sharing the same agent noncooperatively, the manufacturers compete headtohead for the exclusive services of the agent. The salient feature of this model that headtohead competition for exclusive services creates countervailing marketing incentives for the retailer15 The usual incentive for the retailer to understate demand arises to convince the manufacturer that high sales are due to the retailer's diligent efforts, not exogenous demand. This incentive is countered by the following effect of competition. As market size increases, the total value of the retail outlet increases. Hence the manufacturers' bids for the 15For a general discussion of countervailing incentives in principalagent problems, Sappington [1989a,b]. see Lewis and Our model of exclusive agency differs significantly from the model of Biglaiser and Mezzetti [1993]. They examine the optimal labor contract that emerges when two firms compete for a manager whose ability is hidden. do not arise. (1993). Their model is structured so that countervailing incentives Our model of exclusive dealing also differs from GalOr (1991b) and Martimort They consider two manufacturers, each of whom operates through an exclusive dealer. The manufacturers compete through wholesale price rtlwnt.rpam etnrnnstitinn (lalOfr alsn ennsirlrs the es, leaving the resale price to be determined by case of RPML. Instead. we follow Bernheim and retailer's services increase with the perceived market size. The prospect of higher bids can encourage the retailer to exaggerate the size of the market. These countervailing incentives will give rise to partial pooling in the equilibrium contracts. In this competitive scenario, each manufacturer offers the retailer a menu of con tracts (xi(0),pi(), Ti(0)) in exchange for his exclusive services. No wholesale price is necessary since now xi is dictated directly. cepts Mi's Ti(0) is a fixed fee. Once the agent ac contract, he is prohibited from dealing with manufacturer MA, who then has no access to this particular market. Since this is a static model, we assume that a manufacturer can never credibly bid more than the maximum total surplus value of her product. That is, the most that Mi can bid to attract the retailer is 1n,(0) where (xf S(p;(0) mi)xy(O)  e , p) is the outlay that the retailer would choose if the manufacturer's ex clusive contract consisted solely of a wholesale price equal to marginal cost mi. Thus, II,(0) is the profit that the retailer would obtain if he were the full residual claimant of product line i. Equivalently, II(0) is the vertically integrated profit for Mi. Note that the total surplus IIi(O) increases as the market demand 0 increases. For simplicity, we will assume that manufacturer M1 dominates M2 in the sense that II1(0) every 0. II2(0) for This will be the case, for example, if the two product lines are homogeneous and M1 has a lower marginal cost than M2. Alternatively, M1 could dominate M2 if product 1 possesses a stronger brand loyalty than M2's product. Since M1 dominates, we will observe that the retailer always chooses to exclusively serve M1 in equilibrium. Thus, an equilibrium strategy for MA is simply to offer the pooling contract consisting of a wholesale price equal to marginal cost m2, while M1's I~tl rQ inr ro\ V, Pi \VI, 0, 8)~ equilibrium strategy (zl, pl, Ti) solves the following restricted program: [ED]  mxxl( maxp,t, lTl {Tl( u'(e) U(0) )dz such that = x() + p(0o), 112(0), Pi and x1 are nondecreasing in 0, where U(0) = U(0, ) and U(, 0)  i(0)pi(0)  e(x1(0),, p (), 0, )  T1 (O). To facilitate a comparison with the duopoly model, we will continue to assume that e has the general quadratic form given in equation (1). In addition to Assumption 1, we will restrict attention to those Beta distributions with fO nondecreasing in 0. An example is the uniform distribution (A = 1 following characterization of the optimal verti ). With these assumptions we have the cal restraints under competition for an exclusive dealer. Proposition 7 The participation constraint (2} in program [ED] is binding at a unique Moreover, at the solution (xl,pi) to [ED] we have the following 17 the optimal vertical restraint involves resale price Also, p(0)) p;(O) and Xi(0) a;(0) for 0 '6We assume that M/1 wins any ties, that H11 is concave in zi and pi, and that, in equilibrium, the agent will be employed for any realization of demand. how program [ED] is restricted. constraint U(0, 0) However, in most In addition, it is important to understand Note that the optima under the more general program with the > U(0, 0) (instead of constraints (1) and (3)) are not always nondecreasing. cases, these optima are nondecreasing so that program [ED] is indeed the correct program. 1'Since II2(0) is increasing in O, it is not immediately clear for which O's the participation con strain (2) will be binding. However, due to the strict convexity of U(0) that constraint (2) binds only at one point, 9. We assume that H'2(0) (  H2(0), we can conclude E(x(f() + p (), x( ) + pfi ()) for . (1) modified all 0 so that 0 E (6,0), where (zf (), pf ()) solves program [ED] with constraint ;d as U'(6) = z2i() + pl(0) A, where A is sufficiently lai rge so that constraint (2) binds z)}f(z point B . For sluggish markets (0 < 0 floors and quantity forcing. E (8, s> E [8, 8], with equality only at and 0. zero The retailer's informational rents decrease in 0 Moreover, there is an overprovision of promotional effort, For strong markets (0 with efficient effort supplied only at and 9. the optimal vertical restraint entails resale price ceilings and quantity rationing. Also, p1(O) *;e and xi(9) *la x1(O) 0, with equality only at 0 and 6. The retailer's informational rents increase in 0 with zero rents at 0. Moreover, there is an underprovision of marketing effort, with efficient effort supplied only at 0 and 0. Quantity forcing requires that the manufacturer be able to monitor the retailer's level of sales. If such monitoring not possible, the following adjustments in the optimal vertical restraints arise. Corollaru S If the manufacturer cannot monitor the retailer's level of sales, the man ufacturer elevates the price floor The retailer then induced to provide the efficient level of sales for the accompanying inflated price floor. Proposition Corollary clusions reported in Proposition 4 are illustrated in Figures 7 warrant special emphasis. Three con First, price floors can be optimal vertical restraints in sluggish to intermediatesized markets, regardless of whether sales are monitored. Due to the headtohead competition between the man ufacturers, the retailer's reservation utility increases with the market size, giving rise to incentives for the retailer to exaggerate the strength of the market. this incentive To mitigate , the manufacturer calls upon the retailer to generate performance that is particularly difficult to achieve if demand is below its reported level. The elevated price floor forces the retailer to put forth an abovenormal level of marketing inten sity to promote demand at this artificially high price, particularly if demand is low. rents at 0 (wages at 0 are II2(0)) Resale Price Q Monitored Sales FIGURE 41 MONITORED SALES. 59 Second, for an industry in which only exclusive dealers are employed, both price ceilings and price floors should emerge across different markets. price floors should appear in weak markets. We saw above that In contrast, price ceilings arise to limit the retailer's incentive to understate the size of strong markets. Recall that this was the prevailing incentive in the common agency model of Section 2. Thus as in a common agency, this incentive is mitigated by the use of price ceilings and quantity rationing. In summary, we should thus observe both price ceilings and price floors across different markets. This prediction is consistent with the empirical findings of Ippolito 1988]. For example, in 18 resale price maintenance cases involving gasoline retailing, half involved allegations of price floors while the remainder involved price ceilings. Third , rigid retail prices may emerge. For intermediatesized markets, it is op timal for the manufacturer to institute a single, fixed resale price. Recall that the headtohead competition between the manufacturers creates countervailing incen tives for the retailer. The intermediateranged markets in which pooling occurs are essentially those markets in which the retailer has a "conflict" between the incen tive to overstate and understate the size of the market 0. For sluggish markets and booming markets, the price mechanism is sensitive to market demand, continuously increasing in 0. However, the optimal retail price does not vary with the size of the market for intermediatesized markets. This price rigidity can be interpreted as an intertemporal price stickiness. In a dynamic setting, each manufacturer would prefer to commit to the static contract (derived above) in each period (see Laffont and Tirole (1990)). That is, they would prefer to offer the same static contract each period if they could credibly commit to this without renegotiating1s In such a case we would observe intertemporal price rigidities in intermediate markets even when demand is nonstationary. Consequently, Resale Price 0N6 B Unmonitored Sales FIGURE 42 : UNMONITORED SALES Resale Prices When Sales Are Monitored (Complements) (Independent) (Substitutes) Prices: D P duopoly P multiproduct monopoly ED P exclusive dealing L H P, P single good monopoly 0 P Demand FIGURE 43 multiproduct firstbest single good firstbest : COMMON AGENCY VS. EXCLUSIVE AGENCY across time and across markets of considerably different sizes, we may observe the dominant manufacturer instituting the same exact retail price! There is no price discrimination. Note that price stickiness does not arise out of risksharing concerns. The price rigidity is solely the consequence of incentive contracting under manufacturer competition. Fina perfectly competitive price of mi products are homogeneous. lly, note that as Hii() + this constant price approaches the H2(0) for all pooling 0's, if the two These results are due to intense manufacturer competition. In conclusion, present Figure 3 to compare the vertical restraints that emerge under exclusive dealing with those that occur under duopoly. Figure 3 illustrates the predominant case where exclusive dealing results in a lower (higher) resale price when compared with the common agency retailing of complementary (substitute) goods. 4.5 Hospital Exclusive Dealing The above common agency and exclusive dealing models can be reinterpreted to explain hospital competition rather than retailing in general. Consider the manu facturers to be hospitals offering a facility (such as an open heart surgery facility) to a physician group (the agent or retailer). The hospital Mi charges the physician group Ai(xi, pi) for the use of the hospital facility. Here zi is the number of patients that the physicians treat at hospital Mi. The price pi is the fee the physicians charge each of those patients. Moreover, following Chapter 2, physicians can induce patient demand with their unobservable effort, intensity, or quality ei. the healthiness of the patient pool (HMO) that they serve. T High 0 is an unhealthy pool of older patients, The physicians know 'his risk is indexed by while lower 0 is a pool of young, healthy patients. In contrast, the hospitals do not know the HMO's risk factor 0, and so must design incentive contracts to induce the physicians to reveal this infor 63 The physician group than bundles this hospital rate with a physician reimbursement rate and charges this price bundle to the HMO. In particular, when the two hospitals have complementary technologies, they will share the same physician group. The resulting price bundle will offer a discount to the HMO. When hospital technologies are substitutes, the dominant hospital win an exclusive contract with the physician group. The resulting HMO price bundle will now involve "balanced billing" low risk HMOs (e.g., HMOs with young, healthy enrollees). That is , the low risk HMO will be overbilled beyond what a vertically integrated hospital would charge. However, high risk HMOs will be offered a discounted price. Intermediate risk HMOs will be offered a uniform, riskinvariant price bundle. This competitive model may help explain many pricing structures observed in the health care industry. Conclusion In this paper we have derived equilibrium retail contracts for two rival manu facturers competing through a shared retailer who is privately informed about market demand. This scenario is very common in many industries. The equilibrium fran chise taxes we identified are nonlinear, they can be implemented by a menu of linear franchise contracts. characteristics. The duopoly equilibrium has the following additional For substitute (complementary) products, duopolistic manufacturer competition through a common retailer results in higher (lower) price ceilings, more (less) efficient underprovision of retailer promotional effort, and less (more) severe quantity rationing when compared with a multiproduct monopolist. contrast, competition for the exclusive services of the agent results in price floors and an over provision of effort in weak markets. This upward distortion was shown to be due to a nr no\ nmynror rrnTn~n'non ;nrnnfhiloar Par csrnnn ni 2Yn a TnMlt iD RPM a at1 fl 64 Our model predicts that in industries where the rival manufacturers share a com mon retailer (e.g., newspaper distributors, food and beverage distributors, clothing distributors, etc.), only maximum RPM should arise. In contrast, in industries where retailers are exclusive dealers neously in different markets. , both maximum and minimum RPM may arise simulta These are precisely the patterns observed in Ippolito's (1988) study of RPM litigation cases. An additional consequence of manufacturer competition for an exclusive dealer is that the same uniform resale price may be instituted in markets that may differ significantly in size. This, in turn, may lead to an intertemporal price stickiness. The price rigidity emerges in our model because of intense headtohead competition between manufacturers, not because of tacit collusion among manufacturers, as other authors have suggested. A direction for future research would be to investigate the effect that a ban on RPM would have on equilibrium contracts. In particular, it would be interesting to determine whether the final market structure (exclusive vs. common retailer) is affected systematically by a ban on RPM. Moreover, if sticky resale prices do not emerge systematically with a ban on RPM, then our model suggests that resale price rigidity (across markets and across time) may indicate that resale price maintenance is being practiced. CHAPTER 5 CONCLUDING REMARKS Common agency theory deals with a competitive version of principalagent theory in which multiple principals contract with the same agent. This dissertation develops common agency theory for applications to the industrial organization of the health care industry. In the first essay, the efficiency of three health care systems is examined. In a multipayer system the public payer (Medicare) uses a mix of prospective payments and passthrough payments, while the private payer (a managed care insurer) uses a qualitybased reimbursement rate through utilization review. Costshifting in the multipayer system induces the hospital to overinvest in technology. Furthermore, passthrough payments of capital are scaled back in equilibrium since they create a moral hazard problem that allows the hospital to goldplate, i.e, to invest in wasteful, nontechnological capital. passthrough payments. This may explain Medicare's current policy to phaseout In a singlepayer system investment in technology is cur tailed, and goldplating can arise. Whei and the hospital is admitted, technology based reimbursement rate is negotiated. n trilateral negotiations between the payers :al efficiency results when a uniform quality Moreover, this allpayer system is immune to goldplating. The second essay studies the interrelationship between a hospital's capital struc ture and the payment plans designed by Medicare and the private insurance sector to reimburse the hospital's cost of capital and technology. To counter the opportunism of a managed care private insurance sector engaging in utilization review, the hospital will use debt when bankruptcy is costly. The final essay derives the equilibrium hospital contract with a physician group 66 it serves. Bundled physician charges are then derived when hospitals compete for an exclusive contract. Finally, we suggest some theoretical extensions to our model. First, we have considered only deterministic patient demand in the first essay. In reality, the demand for medical services is highly stochastic. In such a case, some range of passthrough payments may be optimal if the hospital is risk averse. Moreover, as shown in the second essay's stochastic model, ex post negotiations may no longer be optimal since they may induce a forprofit hospital to increase its debttoequity ratio, turn increases the hospital's risk of bankruptcy. static model. which in Second, we have only considered a In a dynamic model, insurers may try to behave opportunistically by renegotiating reimbursement rates. Martimort's (1994) model of a dynamic common agency suggests that a multipayer system may be optimal if the hospital has private information concerning its costs. APPENDIX Proof of Proposition 1: The SinglePayer System. First note that the equi tion Ts = 0. So the Lagrangian of the public payer's Ibrium Ts is the boundary program is  aX  pT + pY + AY. The Y> equilibrium 0 constraint will be nonbinding. Thus the first order condition We will later verify that indeed A = 0 provides p = _x ~ XI =Oin Next =UI X XXII  aX + Y + (1 X) Aj Ar =UI cXr X  cX + Y. xryI (Al) Thus . Next we verify that A Therefore crX2 XI = 0. From YI = 0 we have a _ cXl+clX  XI Hence (2) The MultiPayer System. such that for all n < n1, the I First , note that there exists a maximum nl program maxyA has an interior solution solving = (1 n)Ur  crTX +p E (0,1) (A2) For n > ni, the equilibrium To i Lagrangian for the public payer's s the boundary solution To program is = 0. For n =n(U aX) pT + pAt + 7AT + AA. We will for now assume that A = 0 provides p = x XI Thus = 0 and later verify this. S= 0 reveals that The first order condition ATTrr Next nUT from the first order condition X XI = 0 we obtain 7 (A3) when po case, equation (A3) reveals that = TATT + UT  AIT. (A4) Define K(I, T) = TATT + Using equation (A4), equation (A2) becomes AT(I = UT(I, T) cTX(I) nUT(I, T) + nUT(Iro, To) + K(lo, To). (A5) nr..4n 4k.a. ... (F PErF \\l ..j /A\. . (1) solU I/ ,Y E (O, T m, 4V hf,,,e IAE\,,I..,,, L, when po K(Io, To) E (0,1). > 0. First Next we claim since eTT Assumption we have 1 1 (cTTT = IA T  ArT < err) Thus. implies that TATT + IAIT (A7) However, that K(I,T) concavity of X(I), = TATT + we have > 0. Next Thus, equation (A7) , we claim that Ar(Io, TE(o)) implies > 0 (in equation (A6)) when K(lo, To) K( o, To) Then , by concavity, A Suppose instead that AT(Io, TE(Io)) T(lo, TE(Io)) < 0 implies that TE(0o) < 0 when > To(Io). then nUT(Io, To)  nUT(lo, TE(Io)) , implying that K(lo, To) < 0 from equation (A6). when K(lo,To) when K > 0 a This is a contradiction. I, > 0 and po E d Po E (0,1). (0, 1). If po Therefore, we must have AT(lo, TE(Io)) > 0 tat is, the hospital overinvests in technology 1 for all n, then by the intermediate value theorem, there exists an ft E (0,1) such that nUT(Io, TE( o)) < 1 for all n < n. Thus, ArT(o, TE(o))  nUT(Io, TE( o)) for all n when Po < n when po = 1 for all n. a m Now we define no That is as the largest n , the hospital overinvests in technology for n E (0,1) such that (lo(n), To(n)) = (o(n), TE(Io(n))) (A8) From the above analysis, we know that no exists by continuity. To show that over investment in technology occurs for all n only when po = 1. For, using equation note that equation (A8) can hold (A8), equation (A5) can be reduced AT(lo(no),TE(Io(no))) = K since AT = 0 at TE. Thus. that A a = no. P I. Therefore, for n > 0 when po 'o = 1 at no. no, either po S(0,1). land?'o However, this is a contradiction = 1, implying E (0,1) with > TE or Po To_ Next (Al). since 7 Next, r Tb. , we claim that Io When n < nl ai U( a < IE when K(lIo, To) d po (0,1),  aoXI) + PIlAI + PAII + 7AIT = (Ur = T from the first order condition for p. ate that K(I, T) > 0 implies that For n  crX , Io < IE from equation  cX) + xA + TAIr = 0 (A9) Define L(I, T) = J X XI Air + TArT TXIA ATr (A10) and that L(I, T) > 0 implies that Moreover, noUT(Io, TE(lo)) = n(U Th n 69 Thus, multiplying equations (A10) and (All) together provides ATrA1 < AyT. However, this violates the hospital's second order condition ATTAII Ay > 0. Therefore, to satisfy this second order condition, we must have L(I, T) < 0 when K(I, T) > 0 and po E (0, 1). As a result, equation (A9) reveals that BC = L(IE(To), To) 0 when po E (0,1). Thus, the hospital undersupplies intensity. A similar proof holds when po = 1. Finally, we verify that A i) = 0 under Assumption 2 ( t We must show that A(lo, To) > 0 for all n. Define & to be the maximum outofequilibrium a that forces A = 0 for a fixed (I, T) when p = 1. Then n& = c (1 n) while the equilibrium x noo = Cy+c holds if < A. Ux. (3) The AllPayer System. max4YqVXMz is Thus, ao & at (lo, To) if c, 0 and A >(1n)[ K], which = 0 under Assumption 2. The first order condition for the Nash bargaining program qMV = nzVY +(1 n)MY. (A12) Fron equation (A12) we can derive etX = qU + (x + z)(cX + T). Therefore, cX = qU+(x+z  1)(cX + T) = q(U cX T) since q + x + Therefore, = 0 and BY = 0 at (IE, TE). Proof of Proposition 2: (1) SinglePayer System. Note that the public reimbursement plan (a = 0, r = 1,p = 1) would ensure that Y = 0 for any I and T. Since the hospital is indifferent, (IE, TE) is an equilibrium. (2) MultiPayer System. First note that if r > 0, then AT = UT CT X1+p  nUT + nrcTX. (A13) From equation (A2), we see that equation (A13) reduces to AT = nrcTX > 0 at To(I). Thus, T7 > To. To verify that r > 0, we examine the public payer's Lagrangian: n(U aX) nrcX pT + pAI + 7AT + AA. Assume for now that A = 0. The first order condition for a implies that p = Thus, the first order condition for p then provides 7 =Tifp e (0,1). Next = ncX implies that r = (A9) above. Thus nX + [ciX + cXi] + ncTXT Xr 1 in equilibrium. Next, the first order condition on I is equation , (T) = Io(T) for all T when po,pi E (0,1). Proof of Proposition 3: Thus, A(Io, To) The government's program is now maxa,p,G,T n(U aX) p(T Differentiating with respect to a provides p + G) + PAI + 7ArT + AAc. Sx qX' Differentiation with respect to G reveals that A   l~c The first order condition for T is still equation (A3). Next, solving for 7 in equation (A3) and substituting into the first order condition for p when p E (0, 1) provides the equilibrium passthrough payment ATTI/$' ATT + " (nUT + ATI) (A14) Using the the definition of po in equation (A4), equation (A14) reduces to .t, [Po + GATT]. ATT + 4"' (A15) However, since AG = 0 in equilibrium, we must also have p = equilibrium, equation (A15) implies that 1 '(G). Thus, in 4"(G) S+ Po + GArr]TT ATT + "(G) (A16) Letting 4(G) =aG AG2 2 , it is straightforward to show that G* = po b2AsTT > 0 solves equation (A16) and reduces equation (A15) to p =bG* Proof of Proposition 5: Suppose that H = Y= V1' D* = 0. Then z, z = 0. First, we claim that > 0. Let . The private insurer will choose Pf to maximize H by solving the first order condition qV(1 + b4)  q)Y. Next note that 98* HpD OD H0p By concavity of H, H0p < 0. (A17) Moreover, HRY 9Y OV HOD = qV + (1 Vq) 800D aD ap ATT p t G) ~r+ ~I~N(G)  (T + G $  Thus > 0. Hence, Now now that dY I= ( )apO  n)x  bF( z*) + b(1 n) f X~dF. (A18) Since we assume D* ever since , the hospital's we must also have first order condition must be d dD How dY  = ( 9/" n dx from equation (Al D* > 0.O However, this is a contradiction. Therefore, we must have Proof of Theorem 1: We will der ture that A ive the equilibrium form (42) in the following manner. First, 2 takes this form in equilibrium. Then we show that M 's be also takes the quadratic form in (42). To do we conjec st response so, the following Lemma is useful. Lemma A in a market Given the conjecture A2 as defined in (42), thi t of size 6 is U(O) = w2(O) + maxe1,1 (tK(xl,pi e agen ,0) t's equilibrium utility Ai(xi,pi)), where wt(0) = maxxa2,p  02) x2p2   2)p2 72 K(zx,pi,0) = "2x yp + (k3 + (Oki)x1 + (k4 + (02)p1 + ksxip1 for 2 21 = 1 + cti + csiti Ct1(62 + a2sli) ct2(62 + a2S2 =(1 z2) (a2 + 12)(b2 + q?2) (1 h2 a2s)2 (a2 + P2)  O2)22 'a r I r Hpo at + Pt bz + ~2 2 1 rn p, t ([7 1 (8 al + bl $  hl)S fc The proof is tedious, and so is omitted. From the Revelation Principle, for any The details are available upon request. r tax A1 : 3I  R+ that is a best response to the conjecture Az, there exists a direct revelation vertical restraint (pi(0), x1(O), T (0)) such that Ti(0) = A(0a O),pi(0)) when restricted to the domain D = {(xi(0),pi(G)) l o} and (zi(p(0),pi (0), T()) solves the program max, ,x ,TJ / {Ti( u(0, 0) > U(, ), U(, 8) > max{0,i pl, Xl z) mlx (z)}f(z)dz such that w2(0)}, and for all 0,0 u(,0e)  w(O) + + P + (3 + 1)x1i()+ +(k4 + 0),pi() + kx(0)pi(0)  TI(0). It is important to note that the Revelation Principle does not in general hold simul taneously for both manufacturers unless we restrict reports of 0 to 0 (see Martimort and Stole [1993]). Next, let u(0) U(O, ) w2(8) be the rent function corresponding to A1 and suppose that that (xil, p) is a solution to the program [Mi] maxpi,." e {Ti(z) mlxi(z)}f(z)dz such that u'(0) = q1xi(0) + #2p1(0), max{0, w2(0)}, and SP, Xi 0 for all 0,0 Then (xl,pi) will also be an optimum of the program [Mi] iff #1ii(0) + 2zp1(0) is strictly increasing in 0. This conclusion follows from Theorems 7.1 and 7.3 of Fudenberg and Tirole [1991]. Since w2(0) is decreasing and u(0) is increasing, the participation constraint of program [Mi] holds under assumptions (A22) and (A23) below. In fact, if fi 0, i=1,2, then program [Mi] is equivalent to program [M1] if zx and pi are strictly increasing. We will assume a priori that 61 and #2 are nonnegative and then check ez post to see when this is indeed the case at the identified solution. Also, we will assume the following: (A19) (A20) I A a2 + /32 S S > 0, b2 + r2 1) > 0, and (a2 + z)(b2 + ) r( (1 h 2)" = 1,2;  a U , a 1 where EO,   (A23) x1(0)p1(8) e(x1(0),x2(8),p1(0),p2(0),8) mi1 > w2(0 at 0 = where ai(0) and p1(0) are given below and x2(0) and p2(0) optimize wa2 0). Conditions (A19) and (A20) insure that w2(0) is the optimum of a concave pro gram, attained at a positive level of sales. Similarly, [Mi] is a concave program if the inequalities in (A21) hold. Under (A22) and (A23), the agent will be employed by manufacturer M1 for any realization of the demand. Clearly, all five assumptions must be checked ex post since they involve the equilibrium parameters of A2. The Hamiltonian for program [M1] is H(u,p,x,0) = f(O)[IC(x1,pi,0)  u(O) mpx3] + p(0)[iaxx + #2P1], (A24) where u is the state variable. The Pontryagin principle yields /(0) f(0). Further more, O is a free boundary so that p(0) = 0. Integrating provides p(O) Substituting this back into the Hamiltonian, we have = F(9) f()O = ka + x + k3 + 0 + kspl 1 F(O) f()= 0 (A25) f(6) p1 =kzp1 +k4~+6~2 +ksxi. 1 F(0)  / 771 2 (A26) Solving (A25) and (A26) provides xi(6)  p (0) = piW=) z' k" "k2 l ks )  k2kl)  F()\ k52 k k4 m f(O) +k km f (0) k2 k hj 1 F(0) ti + k k4 ,k1t ~+k3 k~Tmln :0) kCg s ks (A27) (A28) and TI(0) = AC(0) fS {t1x1( z) + 2p1( z)}dz  max{0 w2(0)}, where W2(0) _2  ~2)po + (1  h2 a2)xoPo It is immediate from (A27) and (A28) that xz and pi will be increasing in 6 if the inverse hazard rate of f(0) is weakly decreasing when k2qA ks52 < 0 and k51 kq12 > 0. Note that these last two inequalities hold when ks > 0. Finally, from Lemma A, it is evident that k5 > 0 if 1 > h2 +oa, since assumed that 1 > hi (we verify below that 1 = hz + a2 in equilibrium). Next we show that the best response to A2 is of form (42) for all (x,,pi) E V. For the Beta(l, j) density, 1 F() fWe  0). Thus, "lz:  9n 1 at $ P2 bz $ )72 + (8 CY2)50 + (B 8(~2 l)trE4 1)+ IC3 x(e Solving (A29) for 8 provides k2 ) A  +  1+X (A 30) 1+A In terms of pi,  k2kI ks ) 4%)l A  + 1 1+ A (1 + t)(1 1 1 # (A 31) 1+)A (A30) and (A31) can be rewritten AC] (A32) 1 =+[zBp+ A O0 1+ A  BD3. Then T1(O) I1 2 (lyA 2(1 + A)) 2 k2 2 2 ^(^ 2zB 2(1 + A) qS (AO 1+) AC) xl+ hja.+k+ 1  BD) BD pi +ksxipi+ iA AC)  BD)2 ,1(xe AC)  BD) 2yA(1 + A) 2zB(1+ A) 2zB  max{0,w2(0)} (A33) (A32) provides \A AC = (1 + A)_2 yAx_1. that we the tax parameters can be rewritten This allows a simplification of (A33) so as ~1P1, =Ic4+ 2 El', = kl + r, = k2 +r = ks where 14\ (A t\ k ,  kik2  le.,,I 1iI fkAs,  kIlk2  eA = 1 [yAxz + \ + k3+ 1x 6 h ~ kz (1 + ~)((61 SICQ k:2 ~,(xe ~z(Xe ~,e(xe (61(1+~)82 dz(l+ ~)82 ~ Ic~ + (618 7. S e+ 2 1 +A  k4k1x The conclusion that al = ks warrants ks = 1 hi + c2tl. Now note that from then t = 0, resulting in al = ks = 1  that can indeed be sustained in equilibr i=l,2,ifc 2 max{(bi+'1i) (ai+/9i) I by assum Next, the fixed ption (A19). Thus qi we solve for the fixed franchise fees are > 0 at l franchise special Lemma hi. Thu ,ium. FiI i = 1,2 east for e fees. F attention. Recall that by defti A, if we conjecture that Oa = 1 nation  2, s, ao = 1 hi, i=1,2, is a royalty rate nally, note that qi > 0 in equilibrium, } Note that this maximum is negative complements and weak substitutes. rom T1 (0) above, it can be show that 71= + 2 2 ~2P 2 2 7 2 2 if w2(_) > 0. We will show that w2 () > 0 iff c < 0. First, note that 72 is chosen as large as possible by Ms so that as much rent is extracted while still ensuring that the agent's rent in common agency is not below the level of rent the agent would obtain for exclusively serving M1. At 0 this constraint is binding, i.e., max ,2p, {zipi + X2p2 e( 1i,pi,2,2,1P,#) A A} = = max,,,, {zlp e(l, pi, 0,0,0) A1}. (A34) (A34) isolates 72 = maxzzr2p1 {xipi + x2p2 e(Zx,p1,x2,pp2p,) a1 1 6 1l P1 2 71 P2 ai p: oalzpi iax2 62P2 2 2 2 '_22 _ nPaM  max,,, {zpI e(xl,pl,0,0,9) "a li  61P  2 1 2 OlZiPI}. YiPiip} (A35) Refering to Lemma A, we can define Kt1 and KA2 so that t so that KCi is the value of the second maximization in the ( iff Ki + K2 > max x2p{,p^L}, where L is the first maxin Next, we have L < L cxx2 cplp2 iff c < 0. However max1a:21p2m {L 1C2l2 cp1P2}. Therefore, we indeed have KC1 Hence, w2(0) > 0 iff c < 0. Next. for c > 0. suDDose that both narticioation constrain S7 w2(_) = 72 + K2 and A35). Then w2(0) > 0 land in equation (43). , note that XKC + AC2 = +AC2 maxazlp,2 p{L}. nts are binding: 7i = max,2P1p2{L} AK2 (A36) 72 = Then 71  maxxa22pvp{L} KC1. 72 +maxa p2m,1p {L} 'i+ 1K2max,,aap, {L (A37) }. However, the righthand  k5mi1  kik2 .. J C ~ "" ~"r ~"~ """ Finally, we must show that A1 can be extended from VD to 32. with the same quadratic specification (42). For any xz pi E 3R+, let I(x, pi 0) be the tax level for which the retailer's utility from (I(xz, pi ), zi,pi) is u(0). That is, the retailer is indifferent between choosing either (Ai, x (0),pi(0)) and (I(xl,plO),xa,p1), for any xz and pi To show that AR is implementable on S. it suffices to show that I(%a,pi08)< and p1 = pi( can write I(a Al(xl,pi) for all xa, p E + and all 80, with equal 9), where x1(0) and pi(0) are equilibrium choices. "i,Pi 0) = IK () u(O). Then ity only at xl = x From Lemma A (0) . we Ai(zi,pi) I(x,piJl0) zx(0)) + xr1(x(0)  p(0))+p1 Fr(pl(O) 2 pl)+ pi(0)). 2 2 (Pl Note that indeed Al(z1(0),p1(0))  I(ai,piO) = 0. Next, O(A1(xl,Pi)  I(x ,pr0))  0) + ri(z1 However , we claim that 4x* = xl(0). Note that x,1(0)  ) + ;1 _ 1 + rl 1+ P1 l A  1 k12 = ( + P1  k4ks5  k2mi . ) 1+)A +x1 = ~1+ : Similarly, O(A1i(x,pi) at pi(O) I(xal,pI 6) Note also that A > 0, for all (x1,Pi) 1(x,>Pi) I(xl,pilO) # (x(0),pi(0)). 0 is convex. Thus , Al(zl,Pi)  Proof of Proposition 6 and Corollary 3: Let HD and HM be the Hamiltonians for the duopoly and monopoly cases. Then OHD Oxl OHM a2 + #2 (a+ f)z2 +c a+ (0 1 F(8) f(0) OHM OUcD  F(0) f(0) from Lemma A. 1f en However (1 .;nr, .4. IR oW' = OH ari = 0 in equilibrium. SnA irr coamimnnn (A 1 Thus Thb orfnra D aOHM(xl p p  zi)+ for xa* =  k2m1  ~l(s e:(s))+ 92 (88>(pl ~1C8  :1> 8)+:, s41 (e rr  jb~41Cg  El IC2 at + 82 >) o ,\ rFM f ~P \ Next, let HF be the Hamiltonian for the completely informed monopolist. ?= H n) Thus, Oz(pj) > fM, with equality at 8. Also, Then 9HD a9HF 1F(O) + 1fifi Hence, xz < zxF(p ). Similarly, p4' tioning and price ceilings. O Ca a2 + /A < pf (af). 8HiF 1 F() . f(O) Thus, duopoly involves quantity ra Proof of Proposition 7 and Corollary 4: First , note that program [ED] is equivalent to the following program: maxpi, (Pl ml)zl e(xl It z)(xl + pi H' z)dz e(x1 ,O,pi ,O,z) z))(1 i+pi n'2( z)dz such that (1) (2) xl(0)) xl(0) xi(0) z,( ) pdGe) p1(O) for all 8 for all 0 S[, l], E [0,0]. Necessary first order conditions are Gl(xl, pi) = pi  hip +0 + F(9) f(9) = 0 and G2(zx,pi) =a z  6ibP F(0) f(0) E [2,03i (A38) Tn1  alx,  hip + 0 F(0) hllx + 0 F(0) =0 for E ,[3], where i, and r7 are the Lagrangian multipliers for constraint (i) above, i=1,2. > 0 and ri Clearly, on some common nondegenerate interval containing 0 (when E (, l)) or else one of the optimal controls will not be nondecreasing. Finally, to prove Corollary 4, we show that the price floor increases if sales cannot be monitored. 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He entered the doctoral program in the Economics Department at the University of Florida in 1991, and expects to receive a Doctor of Philosophy degree in May 1995. Encinosa has accepted the position of Robert Wood Johnson Scholar in Health Policy at the University of MichiganAnn Arbor. I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David E. M. Sappington, Chairman LanzillottiMcKethan Eminent Scholar and Professor of Economics I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Tracy Lewis James Walters Eminent Scholar of Entrepreneurship and Professor of Economics I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jonathan Hamilton Associate Professor of Economics I certify that I have read this study and that in my opinion it conforms to accept able standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. David Brown Nationsbank Associate Professor of Finance, Insurance, and Real Estate This dissertation was submitted to the Graduate Faculty of the Department of Economics in the College of Business Administration and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1995 L D3 1780 1993 U II I Iiiii iiiii I IIII II lIII i 3 1262 08553 8832 