Inefficient pricing can kill: the case of dialysis industry regulation.

Kaserman, David L.

I. Introduction

Regulation of the health care sector of the U.S. economy is pervasive. Virtually all industries contained in this sector are subjected to myriad government controls over one or more dimensions of performance--price, output, investment, entry, and quality |7; 8~. It is our general thesis that, to varying degrees, the chronic problems exhibited by many individual health care industries can often be traced to ill-conceived, anticompetitive, and conflicting regulatory policies that seriously distort market incentives for firms to operate efficiently. To support this general thesis, we focus here on one particular industry that has recently been shown to perform poorly--the dialysis industry.

Recent empirical evidence strongly suggests that rising mortality rates observed among the population of dialysis patients is at least partially attributable to a pronounced trend of dialysis clinics shortening patients' prescribed treatment duration below medically optimal levels.(1) Here, we show how the current pricing structure imposed on these clinics by the Health Care Financing Administration (HCFA, a part of Medicare) encourages (or, perhaps, even requires) clinics to set running times for dialysis patients at levels below those obtainable with a more efficient pricing structure for the same HCFA expenditure. In addition, we also show how regulatory adjustments to reimbursement rates over time tend to exacerbate this problem. Thus, both static and dynamic aspects of the existing regulatory policy are shown to contribute to this increasingly severe problem. As a result, rising patient mortality appears attributable to an inefficient regulatory pricing policy.

II. Some Facts about the Dialysis Industry

Dialysis clinics earn profits by providing dialysis and related services to persons suffering from renal failure. Eighty-three percent of the independent (i.e., non-hospital based) clinics are operated on a for-profit basis |9~. Their revenues come primarily from HCFA under the End Stage Renal Disease (ESRD) Program, a part of Medicare. This program was initiated in 1972 to relieve kidney patients of the catastrophic costs of dialysis by covering 80 percent of the costs of the service. Expenditures under the ESRD Program have grown phenomenally over time. Its budget has increased from $229 million in its initial year of operation to $3.7 billion in 1988 as the number of patients undergoing dialysis has grown from approximately 11,000 to 110,000 over this period |1; 11~.

The dialysis industry is highly labor intensive. Labor costs, which consist largely of nurses' and technicians' wages, account for some 70-75 percent of total costs. Moreover, this cost structure is dictated by the existing technology for providing dialysis service. Specifically, patients must remain connected to a dialysis machine for approximately two to five hours generally three times per week. This machine performs two essential functions normally provided by the kidneys--it filters impurities from the blood and removes excess fluid.

During treatment, patients must be monitored at regular intervals so that various symptoms that typically arise (e.g., cramps, nausea, and hypotension) can be treated. In addition, all patients must be evaluated (weighed, blood pressure, temperature, and pulse taken, etc.) both prior to and following treatment, and they must be connected to the machine by inserting two large (15 to 17) gauge needles into a vascular access that is usually located in the patient's arm. As a consequence of these care requirements, clinics must employ approximately three to four nurses (RNs and LPNs) or technicians for every ten patients undergoing dialysis treatment at a given time. As a result, the costs associated with these employees per patient dialyzed increase with the duration of the treatment provided.(2)

III. Static Equilibrium under the Current Pricing Structure

HCFA's current regulatory pricing structure for dialysis services consists of a single fixed payment per patient per treatment. At the present time, this payment is approximately $128 on average |4~. It varies slightly from one region of the country to another to reflect cross-sectional differences in nurses' wages, but in all regions it remains a fixed fee per treatment delivered. Because revenues per treatment delivered are unaffected by the length of time the patient is dialyzed under this pricing structure while costs per treatment increase monotonically with the length of run, there is a functional relationship created between profits earned per treatment and the duration of the treatment prescribed. Moreover, as noted above, treatment duration has been shown to be a significant determinant of the efficacy of the dialysis service; i.e., reduced running times cause an increased rate of mortality among dialysis patients, ceteris paribus |3~.

Given this pricing structure, we want to model the dialysis physician's (or clinic operator's) choice of treatment duration. To do so, it is convenient to adopt the following notation and assumptions:

z = patient treatment duration (an indicator of treatment quality);(3)

N(z) = number of patients demanding and receiving dialysis treatment given quality z. We assume N(z) is twice differentiable with N|prime~(z) |is greater than~ 0, N|double prime~(z) |is less than~ 0 for all z;

|C.sub.1~ = fixed costs of dialyzing each patient (i.e., costs that do not vary with the treatment duration);

|C.sub.2~ = constant variable costs per unit of time that the patient undergoes treatment;

p = HCFA's fixed reimbursement rate per dialysis treatment delivered;

|Pi~ = dialysis clinic's profits; and

U(|Pi~, z) = clinic operator's utility as a function of profits and treatment quality (duration). We assume |U.sub.|Pi~~ |is greater than~ 0, |U.sub.|Pi~|Pi~~ |is less than~ 0, |U.sub.z~ |is greater than~ 0, |U.sub.zz~ |is less than~ 0, and |U.sub.|Pi~|Pi~~|U.sub.zz~ - |(|U.sub.|Pi~z~).sup.2~ |is greater than~ 0 (i.e., U(|Pi~, z) is strictly concave).

We assume that the clinic operator selects a level of quality z |is greater than or equal to~ 0 to maximize utility U(|Pi~, z):

|Mathematical Expression Omitted~,

where clinic profit |Pi~ is given by

|Pi~ = p |center dot~ N(z) - |C.sub.1~ |center dot~ N(z) - |C.sub.2~ |center dot~ N(z) |center dot~ z

= N(z)|p - |C.sub.1~ - |C.sub.2~ |center dot~ z~. (2)

The first-order condition for an interior solution, z*, to (1) is given by

|U.sub.|Pi~~ |center dot~ (|Delta~|Pi~/|Delta~z) + |U.sub.z~ = 0, (3)

where subscripts denote differentiation.

We note immediately that (3) requires |Delta~|Pi~/|Delta~z |is less than~ 0 at z*. That is, the clinic operator who cares about quality (i.e., an operator for whom |U.sub.z~ |is greater than~ 0) offers a level of quality beyond that which maximizes profits. The interpretation of condition (3) is straightforward: quality is increased until the utility of the additional profits foregone from further quality improvement (|U.sub.|Pi~~ (|Delta~|Pi~/|Delta~z)) equals the direct effect that quality improvement has on the clinic operator's utility (|U.sub.z~). Because this direct effect is positive (|U.sub.z~ |is greater than~ 0), the quality chosen exceeds that which would result from pure profit maximization.

The second-order condition for z* to solve (1) is given by

|U.sub.|Pi~|Pi~~|(|Delta~|Pi~/|Delta~z).sup.2~ + 2|U.sub.|Pi~z~(|Delta~|Pi~/|Delta~z) + |U.sub.|Pi~~(||Delta~.sup.2~|Pi~/|Delta~|z.sup.2~) + |U.sub.zz~ |is less than or equal to~ 0, (4)

where double subscripts indicate second derivatives. Because U (|Pi~, z) is concave by assumption, and because |Delta~|Pi~/|Delta~z |is less than~ 0 at z* by (3), condition (4) is always satisfied whenever |U.sub.|Pi~z~ is not "too large a negative number" and |Pi~ |is greater than or equal to~ 0. Since one typically expects |U.sub.|Pi~z~ |is greater than or equal to~ 0, the second-order condition for a maximum will generally be satisfied.

The static optimum depicted in equation (3) is illustrated in Figure 1. In this graph, profits appear on the vertical axis and treatment duration (quality) is shown on the horizontal axis. The tradeoff created by the fixed fee reimbursement schedule is shown for a reimbursement rate of |p.sub.0~. This profit-quality frontier first increases in both quality and profits at sufficiently low levels of quality. At very low levels of quality, the number of patients dialyzing at the clinic is sufficiently low that clinic profits are increased by devoting additional resources to increasing treatment quality.

This positive influence of quality on profits may result from two alternative sources. First, where clinics face some competition, low quality may drive patients to dialyze at these other clinics. Second, where quality becomes extremely low, patients must routinely be admitted to the hospital due to complications, or they may die. In either event, sufficiently low quality must lower profits through its effect on the demand for the clinic's services.

Beyond some (profit-maximizing) quality level, |z.sub.0~, however, further increases in quality must come at the expense of profits. Therefore, under the fixed reimbursement schedule, as quality increases, the profit-quality frontier first increases at a decreasing rate, reaches a maximum, and then declines at an increasing rate, because of the effects of z on both costs and N.(4)

The indifference curves of a dialysis clinic operator in |Pi~, z space will be horizontal in Figure 1 if that operator is totally indifferent to the quality of care provided to patients. In that case, the utility maximizing amount of quality will correspond to the profit-maximizing amount at |z.sub.0~. For operators who derive utility from both profits and quality of care, however, indifference curves presumably exhibit a conventional convex shape and negative slope. Such curves are shown as |I.sub.1~, |I.sub.2~, and |I.sub.3~ in the graph. For these operators, utility maximization will occur at some level of quality greater than |z.sub.0~ and some level of profit below ||Pi~.sub.max~. Specifically, with a positive marginal rate of substitution between profits and quality, equilibrium occurs at |Pi~*, z* in the graph.

IV. The Impact of Reimbursement Rates on Quality

We now turn to an analysis of how changes in remuneration rates might affect the equilibrium service quality chosen by dialysis providers under the current pricing structure. While this discussion is a diversion from our main interest (i.e., investigating how a change in the pricing structure could change the quality of dialysis treatment), there are important reasons for an analysis of remuneration rate changes. First, such an analysis aids our understanding of why treatment quality has declined (i.e., mortality has increased) over the 1980s. Second, this discussion of changing rates will show how the rate adjustment process used by HCFA naturally leads to deteriorating quality over time. This, in turn, points to the pressing need for changes in the compensation scheme. Finally, this analysis allows us to investigate whether increasing the reimbursement rate under the current pricing structure would be likely to contribute to quality improvements.

HCFA, as the major payor for dialysis, sets reimbursement rates. The 1980s were a period in which average nominal rates fell substantially. HCFA reduced average reimbursement from $138 to $129 per treatment in 1983, and in 1986 rates were reduced further from $129 to $125. Accounting for general inflation, the decline in real compensation rates was 55% over the last decade.

To show how these changes in reimbursement rates have affected treatment quality, we differentiate (3) with respect to z and p. Rearranging these derivatives and utilizing (4), we conclude that

|Delta~z*/|Delta~p |is greater than~ 0 (5)

when

|U.sub.|Pi~|Pi~~ (|Delta~|Pi~/|Delta~p) (|Delta~|Pi~/|Delta~z) + |U.sub.|Pi~~ (||Delta~.sup.2~|Pi~/|Delta~z|Delta~p) + |U.sub.z|Pi~~ (|Delta~|Pi~/|Delta~p) |is greater than~ 0. (6)

and |Delta~z*/|Delta~p |is less than~ 0 otherwise.(5) Noting that |Delta~|Pi~/|Delta~p = N(z) |is greater than~ 0, |Delta~|Pi~/|Delta~z |is less than~ 0 by (3), and ||Delta~.sup.2~|Pi~/|Delta~p|Delta~z = |N.sub.z~ |is greater than~ 0, it is clear that condition (6) is always satisfied if |U.sub.z|Pi~~ |is greater than or equal to~ 0, and it may be satisfied if |U.sub.|Pi~z~ |is less than~ 0.

In general, one expects condition (6) (and, therefore, (5)) to hold in all but "perverse" cases. To see this, recall that z is a "normal good" in the typical sense whenever |Delta~(|U.sub.z~/|U.sub.|Pi~~)/|Delta~|Pi~ |is greater than~ 0, requiring that |U.sub.z|Pi~~ - (|U.sub.z~/|U.sub.|Pi~~) |U.sub.|Pi~|Pi~~ |is greater than~ 0. Utilizing (2), the first and last terms in (6) can be written as

N (|U.sub.|Pi~|Pi~~ (|Delta~|Pi~/|Delta~z) + |U.sub.|Pi~z~). (7)

Also, by (3),

|Delta~|Pi~/|Delta~z = -(|U.sub.z~/|U.sub.|Pi~~). (8)

Therefore, because it is sufficient for |Delta~z*/|Delta~p |is greater than~ 0 that |U.sub.|Pi~|Pi~~(-|U.sub.z~/|U.sub.|Pi~~) + |U.sub.|Pi~z~ |is greater than~ 0, we conclude that (5) will be satisfied whenever z is a "normal good." Further, though, even if z is "inferior," |Delta~z*/|Delta~p |is greater than~ 0 is still possible so long as z is not "extremely inferior." Therefore, it seems very likely that |Delta~z*/|Delta~p |is greater than~ 0 for most (if not all) dialysis clinics. That is, an increase in the fixed fee reimbursement schedule will, in all likelihood, contribute to an improvement in the quality of care (treatment duration) in this industry.

The typical "normal good" case is depicted in Figure 2. Here |p.sub.0~ is the quality-profit tradeoff with reimbursement rates fixed at |p.sub.0~. Increasing reimbursement to |p.sub.1~ shifts this frontier upward and to the right.(6) As a result, equilibrium shifts from |z*.sub.0~, ||Pi~*.sub.0~ to |z*.sub.1~, ||Pi~*.sub.1~. Increased funding under the current pricing structure does lead to quality (and profitability) improvement in the "typical" case.

The policy implications of the analysis above are straightforward. First, it seems likely that the current quality problems exhibited by the dialysis industry have been exacerbated by the reductions in reimbursement rates instituted during the 1980s. Further, it seems likely that, under the current pricing structure, quality could be improved by raising reimbursement levels. What remains is to find if another pricing structure can achieve these quality improvements more efficiently. Before we address this question, however, we explore the dynamic problems created by the current pricing policy.

V. Dynamic Problems under the Current System

The current system through which HCFA regulates dialysis clinics' reimbursement rates exhibits two fundamental shortcomings of a dynamic nature. These shortcomings contribute to the tendency for quality of care to decline over time. First, HCFA adjusts its reimbursement rates from time to time solely on the basis of observed profits (or costs). Periodic audits of dialysis clinics' costs are conducted, and funding levels are altered to bring reimbursement in line with these observed costs. The quality of the service being rendered (in particular, treatment duration) is not considered as a factor in this regulatory price adjustment process.

The inevitable result of this singular focus on profitability in setting reimbursement rates is shown in Figure 3. Initially, reimbursement is set at |p.sub.0~ per treatment, leading to an equilibrium at point a, with profits and quality at ||Pi~*.sub.0~ and |z*.sub.0~, respectively. Let normal profits be given by |Mathematical Expression Omitted~ in the graph. Upon auditing this clinic's costs, HCFA finds that it is earning excessive profits equal to |Mathematical Expression Omitted~. To lower profits to a normal level with quality maintained at |z*.sub.0~, HCFA reduces reimbursement rates to |p.sub.1~, anticipating a new equilibrium at point b. Facing the profits-quality tradeoff of |p.sub.1~, however, the clinic lowers the quality of care provided (mainly through reduced treatment duration) and locates at the new equilibrium point c, with profits and quality equal to ||Pi~*.sub.1~ and |z*.sub.1~, respectively.

Subsequently, HCFA audits the clinic's costs again and again finds excessive earnings (equal to |Mathematical Expression Omitted~). As a result, it lowers reimbursement rates once more. This sequential process of adjustment and readjustment by the regulatory agency and the dialysis clinics continues until the stationary equilibrium at point d is attained.(7) Here, the clinic earns normal profits, but quality has been driven to |z*.sub.n~, well below its initial level. Assuming that the initial quality provided by the dialysis industry was approximately equal to some socially desirable level, the dynamic process through which reimbursement rates are adjusted ensures that quality will be driven to socially sub-optimal levels. Thus, this myopic regulatory rate adjustment methodology contains a built-in mechanism that causes treatment quality to deteriorate over time.(8)

The second dynamic process that serves to further exacerbate the quality problem in this industry has to do with incentives created by the current pricing structure for ownership of dialysis clinics to change over time in a direction that contributes to the problem of declining quality. Because of the profits-quality tradeoff created by the fixed fee reimbursement schedule, the assets embodied in a dialysis clinic are worth more (i.e., yield higher profits) to those individuals that are relatively more willing to trade off quality for higher profits. That is, for a given reimbursement level, equilibrium profit is higher the lower the marginal rate of substitution of quality for profits.

This result is shown in Figure 4. Here, individual A has a relatively strong preference for quality. Given the profits-quality tradeoff for a reimbursement rate of |P.sub.0~, equilibrium occurs at point a, yielding a profit, quality combination of ||Pi~*.sub.A~, |Z*.sub.A~. Alternatively, individual B has a relatively weak preference for quality. As a result, given the same reimbursement schedule, this individual locates at equilibrium point b, yielding a profit, quality combination of ||Pi~*.sub.B~, |Z*.sub.B~. Because ||Pi~*.sub.B~ |is greater than~ ||Pi~*.sub.A~, the clinic's assets are worth more to individual B, and market forces will tend to reallocate these assets to that individual.

This market incentive for dialysis industry assets to migrate to owners with relatively weak preferences for quality of care explains another trend being observed in this industry--an increasing number of clinics are being purchased by relatively large health care companies from the individual physicians who originally opened them.(9) Because these companies are generally managed nationally, the consequences of lower quality are not immediately apparent to the new owners.(10) Thus, a market-driven process of self-selection of dialysis clinic owners exists under the current funding methodology that drives industry assets into the hands of those most willing to sacrifice quality for profits and thereby further contributes to declining quality of care over time.

In combination, the two dynamic processes described above (the HCFA price adjustment mechanism and the incentive for ownership to change to those who value quality least) will tend to push quality toward the pure profit-maximizing level (i.e., the maximum point on the profits-quality tradeoff). Since this is the point that is obtained when the clinic operator values quality of care at zero (i.e., is indifferent to quality), the long-run equilibrium quality under this policy will necessarily be below the socially desirable level if society places positive value on the quality of care received by dialysis patients. Thus, the current reimbursement methodology is inconsistent with achievement of a socially desirable level of quality in this industry.

VI. A More Efficient Pricing Structure

One of the fundamental principles of efficient pricing is that costs should be reimbursed in the same fashion they are incurred. This is the basic principle of cost-causative pricing. In the dialysis industry, application of this principle implies that clinics should receive reimbursement under a two-part tariff, because each patient dialyzed causes two types of costs to be incurred. First, there are the costs associated with initiation and termination of the treatment that do not vary with the length of time each patient is dialyzed, and second, there are the costs associated with the duration of each treatment. With a two-part tariff, clinics receive a per-patient treatment payment of |p.sub.1~ regardless of prescribed running time, z, in addition to a second payment of |p.sub.2~z that depends directly on treatment duration (or quality). If, for example, a given patient is given a treatment duration of z, clinic compensation is |p.sub.1~ + |p.sub.2~z.

Under the two-part tariff, clinic profit |Pi~ is given by

|Pi~ = N(z)||p.sub.1~ - |C.sub.1~~ + N(z) |center dot~ z||p.sub.2~ - |C.sub.2~~. (9)

Consider the problem max U(|Pi~, z) s.t. z |is greater than or equal to~ 0 where |Pi~ is as defined in (9). In this case, optimal quality z* will depend on both |p.sub.1~, per patient compensation, and |p.sub.2~, payment per unit of treatment time. In order to determine whether a two-part tariff exists that (1) does not lead to diminished quality, (2) saves money in program costs, and (3) does not require any additional monitoring or information on the part of the program administrative agency, we evaluate the following conceptual experiment. Begin by assuming |p.sub.1~ = p and |p.sub.2~ = 0, i.e., the two-part tariff is selected so that it is initially identical to the uniform per treatment compensation scheme examined earlier. Next, change |p.sub.1~ and |p.sub.2~ simultaneously by lowering |p.sub.1~ (d|p.sub.1~ |is less than~ 0) and raising |p.sub.2~ (d|p.sub.2~ |is greater than~ 0) so that the clinics' desired level of quality z* does not change. If |p.sub.1~ and |p.sub.2~ are adjusted in this manner, then total patients served N(z*) and treatment quality z* will remain exactly as they were under the uniform payment scheme. Program costs, however, will not remain the same. We will show, in fact, that these total program costs must fall in all but "perverse" cases.

Performing the necessary differentiation, the ratio of decreases in |p.sub.1~ to increases in |p.sub.2~ that keep quality the same is given by:

(|Delta~|p.sub.1~/|Delta~|p.sub.2~) = - (|U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~z) |center dot~ N |center dot~ z + |U.sub.|Pi~~ (N|prime~ |center dot~ z + N) + |U.sub.|Pi~z~ |center dot~ N |center dot~ z) |center dot~ |(|U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~z) |center dot~ N + |U.sub.|Pi~~ |center dot~ N|prime~ + |U.sub.|Pi~z~ |center dot~ N).sup.-1~ = - z* - |U.sub.|Pi~~ |center dot~ N|(|U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~z) |center dot~ N + |U.sub.|Pi~~ |center dot~ N|prime~ + |U.sub.|Pi~z~ |center dot~ N).sup.-1~. (10)

Program total expenditure (exp) is

exp = N(z*)|p.sub.1~ + N(z*)z*|p.sub.2~ = N(|p.sub.1~ + z*|p.sub.2~). (11)

Since z* is held constant by the price changes, the change in HCFA expenditures |Delta~exp from altering prices in this manner is only

|Delta~exp = N(d|p.sub.1~ + z*d|p.sub.2~). (12)

Applying condition (10) to condition (12) allows us to conclude that

|Delta~exp = N(|U.sub.|Pi~|Pi~~ |center dot~ N)|(|U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~z) |center dot~ N + |U.sub.|Pi~~ |center dot~ N|prime~ + |U.sub.|Pi~z~ |center dot~ N).sup.-1~ d|p.sub.2~, (13)

which is negative (recall that d|p.sub.2~ |is greater than~ 0) in nonperverse cases. Hence, except in the perverse case for which an increase in prices causes a decrease in quality (which requires that quality be highly inferior), program costs can always be reduced by replacing a uniform pricing scheme with a two-part tariff while still achieving the same treatment quality. Conversely, program costs can be held constant while quality is improved. Or, of course, some mixture of both lower costs and higher quality may be achieved.

The policy implications of these results are obvious. First, the current reimbursement structure for dialysis is clearly socially non-optimal. Resources are wasted and lives are needlessly lost by continued use of a myopic single payment per patient per treatment. Second, adoption of a two-part reimbursement schedule could reduce costs, improve quality, or achieve a combination of both effects.

Further, a two-part tariff of the kind we propose is clearly administratively feasible. Detailed records are currently kept on the duration of patients' treatment. Hence, basing part of the payment made to dialysis providers on treatment duration does not require data beyond that currently collected. We therefore urge the implementation of a two-part tariff for dialysis reimbursement designed to achieve increased quality (treatment run time) which, as we have shown, need not involve additional program costs.

VII. Conclusion

The relationship between treatment duration and the health of patients undergoing hemodialysis is now well documented. It appears beyond question that declining treatment duration over recent years has contributed to increased mortality rates among dialysis patients. Similarly, as economists, it also seems to us beyond question that when compensation for dialysis is a fixed rate per patient per treatment, clinic operators will, within limits, avail themselves of opportunities to trade off treatment quality (treatment duration) for additional profits. Furthermore, this problem is very easy to fix. All that is needed is a two-part tariff that contains a fixed component per treatment delivered and an additional component that varies directly with treatment duration.

Given the preponderance of evidence regarding the virtues of this two-part tariff (versus the current single rate per treatment) one might expect a ground swell of support for change. Unfortunately, those charged with making such changes are medical practitioners who tend to support retention of the current payment format with an increase in reimbursement rates. While we do not expect medical practitioners to be exempt from rent seeking incentives, we must decry the suffering, loss of life, and wasted resources that result from something so easily corrected.

Moreover, we strongly suspect that the dialysis industry is not unique in this regard. Many health care markets are subject to a complex web of ill-conceived and conflicting regulatory policies that, no doubt, contribute to mounting cost and quality problems.(11) A comprehensive rationalization of medical regulatory policy based on efficient pricing principles should rank high on the list of needed health care reforms.

Appendix

Derivation of results (5) and (6):

Let z* solve max U(|Pi~, z), z* |is greater than~ 0. Then z* solves

|U.sub.|Pi~~(|Delta~|Pi~/|Delta~z) + |U.sub.z~ = 0 (i)

and

|U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~z) + 2|U.sub.|Pi~z~(|Delta~|Pi~/|Delta~z) + |U.sub.|Pi~~(||Delta~.sup.2~|Pi~/|Delta~|z.sup.2~) + |U.sub.zz~ |is less than or equal to~ 0. (ii)

Our goal is to derive a condition under which |Delta~z*/|Delta~p |is greater than~ 0.

Differentiate (i) totally with respect to z and p to obtain

(|d.sup.2~U/d|z.sup.2~)dz + (|U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~p)(|Delta~|Pi~/|Delta~z) + |U.sub.|Pi~~(||Delta~.sup.2~|Pi~/|Delta~p|Delta~z) + |U.sub.|Pi~z~(|Delta~|Pi~/|Delta~p))dp = 0 (iii)

at z*. Note that (|d.sup.2~U/d|z.sup.2~) is the expression in (ii) and is negative for an interior z*.

Rearrange (iii) to obtain

|Delta~z*/|Delta~p = -||U.sub.|Pi~|Pi~~(|Delta~|Pi~/|Delta~p)(|Delta~|Pi~/|Delta~z) + |U.sub.|Pi~~(||Delta~.sup.2~|Pi~/|Delta~p|Delta~z) + |U.sub.|Pi~z~(|Delta~|Pi~/dp)~|(|d.sup.2~U/d|z.sup.2~).sup.-1~. (iv)

1. The most comprehensive study dealing with treatment duration of which we are aware is that by Held, et al., |3~. In this HCFA funded study the authors utilize the substantial data available from HCFA to document the strong correlation between decreasing treatment duration for dialysis patients and the increase in patient mortality, holding case mix constant. For other work on treatment duration and patient mortality see Held, et al. |2~ and Lowrie and Lew |5~. Additional information on dialysis reimbursement rates can be found in Kaserman |4~.

2. Capital costs also increase with treatment duration because additional machines and floor space may be required to dialyze a fixed number of patients more hours per week. Thus, virtually all inputs must increase with increases in treatment duration.

3. Obviously, numerous other factors influence the quality of the treatment received (e.g., the nurse-to-patient ratio, the vintage of the machines, the blood flow rate, the size of the dialyzer, the physical surroundings, etc.). We focus on treatment duration here because that appears to be the primary endogenous determinant of quality and the one most directly affected by the current regulatory pricing structure. One may easily generalize our results simply by thinking of z as a composite measure of all quality determinants.

4. This result may be seen by inspection of

|Delta~|Pi~/|Delta~z = |N.sub.z~(p - |C.sub.1~ - |C.sub.2~z) - N|C.sub.2~.

When z becomes large enough, |Delta~|Pi~/|Delta~z |is less than~ 0.

5. See the Appendix for derivation of inequalities (5) and (6).

6. This result may be seen by differentiating |Delta~|Pi~/|Delta~z from footnote 4, above, with respect to p. Doing so yields

||Delta~.sup.2~|Pi~/|Delta~z|Delta~p = |N.sub.z~ |is greater than~ 0.

Thus, the slope of the profits-quality tradeoff increases at every z as p is increased.

7. Note, however, that there is nothing inherent in this problem that guarantees such a stationary equilibrium exists.

8. HCFA has also used observed entry to signal the adequacy of profitability in this industry. This procedure, however, is equally flawed. In the presence of endogenous quality variation, the only thing that observed entry signals is that someone can earn positive profits by providing a sufficiently low level of quality. Consequently, observed entry (or even the absence of observed exit) cannot be used as a signal that profits or reimbursement rates are at or above the socially optimal level.

9. For example, for-profit dialysis facilities as a percentage of all dialysis facilities (both independent and hospital-based) increased from 38% in 1982 to 55% in 1990. See United States Renal Data System |10~.

10. The physicians employed by dialysis clinics still retain responsibility for writing patients' prescriptions for treatment duration under the new ownership. Opportunities for significant quality reductions nonetheless remain because (a) management controls most of the other quality determinants (the nurse-to-patient ratio, the quality and age of the dialysis machines, etc); and (b) management may be able to hire physicians who are relatively more willing to sacrifice quality for profits. This line of reasoning suggests an interesting empirical study of the impact of dialysis clinic ownership (physician versus non-physician) on the quality of care delivered.

11. For example, see Mortenson |6~ for some parallels to drug regulation in a chemotherapy setting.

References

1. Fisher, Anne B., "Washington Reins in the Dialysis Business." Fortune, July 1983, 66-69.

2. Held, Philip J., Jose R. Gariea, Mark V. Pauly, and Marjorie Cahn, "Price of Dialysis, Unit Staffing, and Length of Dialysis Treatments." American Journal of Kidney Diseases, May 1990, 441-50.

3. Held, Philip J., Nathan W. Levin, Randall R. Borbjerg, Mark V. Pauly, and Louis H. Diamond, "Mortality and Duration of Hemodialysis Treatment." Journal of the American Medical Association, February 1991, 871-75.

4. Kaserman, David L., "Reimbursement Rates and Quality of Care in the Dialysis Industry: A Policy Discussion." Issues in Law and Medicine, Summer 1992, 81-102.

5. Lowrie, Edmund G. and N. L. Lew, "Death Risk in Hemodialysis Patients: The Predictive Value of Commonly Measured Variables and an Evaluation of Death Rate Difference Between Facilities." American Journal of Kidney Diseases, May 1990, 458-81.

6. Mortenson, Lee, "Public Policy and Access to New Drugs: The Case of Cancer Chemotherapy," in The Changing Economics of Medical Technology, edited by Annetine C. Gelijns and Ethan A. Halm. Washington, D.C.: National Academy Press, 1991.

7. Phelps, Charles E. Health Economics, New York: Harper Collins Publishers, 1992.

8. Sloan, Frank A., "Regulation and the Rising Cost of Hospital Care." Review of Economics and Statistics, November 1981, 479-87.

9. United States Department of Health and Human Services. Health Care Financing Research Report: End Stage Renal Disease, 1988. HCFA Publication No. 03299, Baltimore: September 1990.

10. United States Renal Data System. 1991 Annual Data Report. The National Institutes of Health, The National Institute of Diabetes and Digestive and Kidney Diseases, Division of Kidney, Urologic, and Hematologic Diseases, Bethesada, Maryland, August 1991.

11. Winslow, Ron, "Cost Control May Harm Dialysis Patients." Wall Street Journal, February 1991. B1, C4.