Scope, learning, and cross-subsidy: organ transplants in a multi-division hospital - an extension.

• Author: Barnett, A.H.

1. Introduction

   In a recent paper in this Journal, Possai and Goetz [10] (P-G) offer an explanation of why a hospital might choose to provide a service (organ transplants) or continue to provide a service that, on its own merits, may appear to be unprofitable. Their explanation is based upon hypothesized (and plausible) cost and demand relationships involving economies of scope, demand complementarities, and learning-by-doing that render the net effects of the service profitable for the hospital's overall operations while appearing unprofitable on its own. Thus, the (undocumented) perception that the transplant business is unprofitable is an illusion caused by a failure to consider the full range of effects of a transplant program on costs and demands, both for transplants and other hospital outputs.

   There are two fundamental and interrelated issues that P-G do not address. First, the analysis they present does not explain why hospitals that do not currently have organ transplant programs are rapidly entering this industry. Nor does it explain why hospitals that already have transplant programs are actively trying to expand the number of operations beyond that presently performed. Rather, P-G's analysis focuses on the static optimality (first-order) conditions for the four objective functions considered and describes the implications of those conditions for overall, firm-wide profitability versus service profitability. Thus, the explanation provided is one of profitability of a service that, on a more narrow accounting, appears unprofitable. Assuming the hospital is already at these optimal values (and there is no reason given why they are not), however, there is no obvious incentive for either entry or expansion, which are dynamic phenomena, to occur. The analysis we present here will more directly address these issues.

   The second fundamental issue not addressed in P-G's approach is the implications for observed behavior in this industry of the persistent and severe shortage of human organs for transplantation. Both federal and state laws proscribe the purchase or sale of human organs for transplantation.(1) Such organs may be donated, but they may not be sold. As a result, the legally mandated price of this crucial input is fixed at zero, and a chronic organ shortage has plagued the industry since transplants first became feasible in the mid-1950s. That shortage has become increasingly severe in recent years.(2) It is difficult to model equilibrium conditions in this market without recognizing and incorporating the effects of this shortage on observed outcomes. Indeed, it is our thesis here that one cannot understand the underlying incentives to enter and expand in this business without first taking account of the organ shortage.

2. A Simple Model of an Input-Constrained Firm

   Over the years, economists have modeled the behavior of firms subject to a variety of exogenous constraints.(3) To our knowledge, however, the particular constraint examined here - where an input price is held fixed below the equilibrium level - has not yet been explored systematically. Such a constraint is clearly relevant to an analysis of firms (hospitals) supplying human organ transplants due to the legal prohibition on purchases and sales of the organs required for such transplants. Accordingly, we begin by developing a simple model that, we believe, reflects the essential effects of this institutional feature of the transplant industry.

   Assumptions

   Our model makes use of the following set of notation and assumptions:

   1. The price of transplantable organs, [P.sub.O], is legally fixed at zero.

   2. Transplant providers receive a fixed rate, [P.sub.T], per transplant performed, paid by a third party. Recipients have a willingness to pay for transplants, but cannot be individually billed for procedures.

   3. The production function for transplant operations is characterized by a fixed-proportions technology in which each operation requires one transplantable organ. Thus, we assume that [Q.sub.T] = [Q.sub.O], where [Q.sub.T] is the number of transplant operations performed and [Q.sub.O] is the number of transplantable organs acquired by the transplanting hospital.

   4. The marginal cost of performing a transplant operation is constant at [MC.sub.T].

   5. The supply curve of organs is given by [Q.sub.O]([P.sub.O]), with d[Q.sub.O]/d[P.sub.O] [greater than or equal to] 0, and [Mathematical Expression Omitted]. Thus, the organ supply curve is coincident with the [Q.sub.O] axis up to [Mathematical Expression Omitted], and it increases in [P.sub.O] thereafter.

   6. Hospitals have significant monopsony power in the procurement of organs within their respective geographic collection regions.

   A brief explanation of these assumptions may be useful. The first assumption reflects the fact that, under the National Organ Transplant Act of 1984, it is a felony to buy or sell human organs for purpose of transplantation. The second assumption captures the exogeneity of reimbursement rates for transplant procedures. These rates are set administratively either by funding agencies (e.g., the Health Care Financing Administration fully funds all kidney transplants under the End Stage Renal Disease Program) or by insurance companies and are, therefore, exogenous to the individual hospital.(4) Further, in an effort to reduce recipient bidding for transplants, guidelines prohibit additional direct charges by hospitals to patients requesting or receiving transplants, although some violations of these restrictions have apparently occurred. The third and fourth assumptions provide a reasonably accurate depiction of the transplant production process and also serve to substantially simplify...