Abstract
We study the effects of 'balance billing', i.e., allowing physicians to charge a fee from patients in addition to the fee paid by Medicare. First, we show that on pure efficiency grounds the optimal Medicare fee under balance billing is zero. An active Medicare policy thus can only be justified when distributional concerns are accounted for. Extending the analysis by Glazer and McGuire, we therefore analyze the optimal policy from the patients' point of view. We demonstrate that, from the patients' perspective, a positive fee can be superior under balance billing. Furthermore, patient welfare can be lower if balance billing is prohibited. In particular, this is the case if the administrative costs of Medicare are large. However, we cannot rule out that prohibiting balance billing may be superior. Finally, we show that payer fee discrimination increases patient welfare if Medicare's administrative costs are high or if Medicare's optimal fee under balance billing implies lower quality for feeonly patients.
JELclassification: I11, I18, H51
Keywords:
physician reimbursement; price controls; Medicare1 Introduction
The US Medicare program allows doctors to 'balance bill' patients, i.e., to charge them a price in addition to the Medicare payment. In the late 80s and early 90s, state and federal legislation was introduced to restrict this practice. Additional prices are now limited to about 10% of the Medicare fee.(endnote a) In a theoretical study, Glazer and McGuire have shown that these restrictions on balance billing come at a price as doctors have an incentive to reduce the quality of their services [1]. Strikingly, prohibiting balance billing reduces quality for all patients, regardless of whether they pay a balance bill. From an efficiency point of view, they demonstrate that allowing balance billing always leads to superior results if the Medicare fee is set appropriately.
A limitation of the analysis by Glazer and McGuire is that they focus exclusively on the efficiency aspects of balance billing. An important concern, however, is that patients are worse off if physicians are allowed to balance bill. In particular, previous work by Paringer, Mitchell and Cromwell as well as Zuckerman and Holahan has shown that allowing physicians to charge extra fees may only increase the rents of physicians at the expense of patients [24]. These papers, however, do not consider effects on quality. Taking into account efficiency gains from balance billing, this raises the question on how these gains are shared between patients and physicians.
In this paper, we take the analysis of Glazer and McGuire further and focus on the welfare of patients. We analyze the optimal Medicare fee both from a pure efficiency perspective and from the patients' point of view. Furthermore, we reexamine the case for prohibiting balance billing and consider the effects on patient welfare if Medicare discriminates the fee depending on whether the physician treats the patient at the fee only or charges a balance bill.
The paper proceeds as follows. In Section 2, we discuss the literature. Section 3 reviews the analysis by Glazer and McGuire. In Section 4, we determine the optimal Medicare fee under balance billing using the social surplus function of Glazer and McGuire. Section 5 analyzes the implications of Medicare's policy on patient welfare. Section 6 concludes the paper.
2 Review of the literature
Most of the theoretical studies on balance billing assume a monopolistic physician who faces a downwardsloping demand curve [2][4]. Within this framework, the effects on the quantity of services supplied by the physician has been explored. The physician is able to price discriminate, requiring patients with a high willingness to pay a balance bill. If the physician also accepts feeonly patients under balance billing, then prohibiting balance billing leaves the quantity of supply unchanged since only inframarginal patients are balance billed. Only the physician's rent is reduced. However, if doctors refuse to treat feeonly patients under balance billing, then prohibiting balance billing reduces the number of patients treated.
How Medicare's balance billing policy affects the incentives for a monopolistic physician to set quality of treatment is analyzed by Feldman and Sloan as well as Wedig, Mitchell and Cromwell [5,6]. Both papers assume that the physician is not able to price or quality discriminate. Feldman and Sloan show that it is uncertain whether price controls, i.e., prohibiting balance billing, increase welfare. Wedig et al., however, find a case for price controls if health insurance shifts the demand curve to the right and physicians react by increasing quantity and quality beyond the social optimum.
All the models presented do not include competition among physicians. Furthermore, neither Feldman and Sloan nor Wedig et al. consider price and quality discrimination. However, these factors are highly relevant in the context of balance billing. First, Medicare's fee policy affects the degree of competition between physicians. Second, balance billed patients are likely to receive higher quality than feeonly patients. Both factors are incorporated in the model by Glazer and McGuire. They show that physicians have an incentive to save costs by reducing quality for Medicare patients. To patients who pay a balance bill, however, they will provide the efficient quality level. Their main result is that by setting fees correctly, efficiency is higher if balance billing is allowed.
An empirical study of the effects of Medicare restrictions on balance billing in late 80s and early 90s has been performed by McKnight [7]. She finds that these reduced outofpocket medical expenditure of Medicare beneficiaries by 9%. With the exception of a significant fall in the number of followup telephone calls, her study shows little evidence that physicians changed their behavior in response to the balance billing restrictions.
3 The analysis by Glazer and McGuire
3.1 The model
In the model by Glazer and McGuire, patients demand one unit of service per period and are uniformly distributed on a line segment of length one. The two physicians are situated at the end points of this segment. A patient's distance from a physician captures the product differentiation which implies that each supplier faces a downwardsloping demand curve. It serves as a geographic metaphor for patients' preferences for treatment.
Demand results endogenously from the benefit
Glazer and McGuire show that a profit maximizing physician always sets the service
quality for the pricepaying patients on the efficient level determined by v'(s*) = 1. This is due to the additive specification of the utility
3.2 Market equilibrium
With the above assumptions, the physicians' demand functions can be derived. Let
When discriminating patients, physicians' are limited by their patients' option to
go to the other physician. They will only be willing to pay p_{i }if this is superior to seeking treatment from the other physician with quality s_{j }at the fee only. For the indifferent pricepaying patient with distance
This shows that the discrimination rule of physicians is based on an endogenous limit between feeonly and balancebilled patients.
Analogously, the total number of i's patients follows from the indifference
Thus, the number of i's feeonly patients amounts to
Using (1) and (3), physician i's profit can be written as a function of her strategy vector (p_{i}, s_{i}) and of her competitor's one:
The market equilibrium is a Nash equilibrium of the complete information game where
1. the two physicians simultaneously choose the price p for the pricepaying and the quality s for the feeonly patients, and
2. each patient chooses one physician.
This market equilibrium yields an endogenous number of balancedbilled patients
The necessary conditions for such an equilibrium are given by
Assuming that the secondorder conditions are met, Glazer and McGuire confine their analysis to symmetric and stable equilibria (see Figure 1), whereby stability requires the slope of the reaction functions in the equilibrium to be smaller than one. This implies for s_{i }= s_{j }= s and p_{i }= p_{j }= p
Figure 1. Symmetric equilibrium in the market for physician services.
Based on this condition, it is possible to show that in the symmetric Nash equilibrium quality increases with the fee f. Accounting for symmetry, (4) and (5) can be rewritten as
Substituting p = (1 + s + v(s))/2 from (7) in (8) yields
By implicit differentiation of (9),
because of (6). Physicians compensate for the lower fee by cost savings from reduced quality for the feeonly patients. Furthermore note that (7) implies
i.e., physicians reduce the price for higher quality if Medicare raises its fee.
3.3 Welfare analysis based on efficiency
In the symmetric market equilibrium, both the quality s for the feeonly patients and the price p for the pricepayers depend on the fee f (cf. conditions (7) and (8)). The model can therefore be used to characterize the socially optimal fee f^{p }if price and quality discrimination are allowed. Because patients split evenly between physicians in the symmetric equilibrium independently from f, the preference parameter t is not relevant for this analysis. Glazer and McGuire consider a social surplus function where the price p as a pure transfer from patients to physicians does not enter. Concerning efficiency, only quality hence remains crucial and total welfare with price discrimination can be written as
Here, v(s)  s measures the net social gain of quality s per unit of service. θf corresponds to the social cost of Medicare, where θ >0 indicates positive administrative costs.
The net social gain vanishes for s = 0 and is negative for all other values of s due to v(0) = 0, v'(0) = 1 and the concavity of v. In a firstbest world, the fee f would therefore be chosen such that s = 0 for all patients. This level is denoted by f*. By equation (9), we obtain f* = 1/2 + c.
With positive social cost θf, Glazer and McGuire identify a tradeoff between quality and distortion costs if Medicare cannot dictate the level of quality and set s = 0 and f = 0 at the same time. Using equation (10), they show that the secondbest optimal fee is f^{p }< f*, implying s(f^{p}) >0, i.e., the secondbest quality received by the feeonly patients is lower than that of the pricepaying patients.
Glazer and McGuire compare the welfare level under price and quality discrimination
and fee policy f^{p }with the situation where discrimination is prohibited. As shown in the preceding section,
total demand of physician i then consists only of feeonly patients whose number is
In the equilibrium without discrimination, social welfare is
since all patients get the same quality s(f) determined by (12) and the total number of patients is one. The optimal fee f^{o }under this regime hence is defined as argmax _{f }W^{o}(f) and solves
Differentiating (12) yields ds/df = 1/(v″(s)  v'(s)) <0. The optimality condition (13) hence can only be satisfied for θ >0 if v'(s(f^{o})) <1. This implies s(f^{o}) >0 because v'(0) = 1 and v″ <0. Hence, if balancebilling is not allowed and Medicare pays the optimal fee f^{o }all patients receive a service of suboptimal quality compared to the firstbest. Glazer and McGuire are able to show that welfare with no discrimination and the optimal fee f^{o }is always lower than welfare resulting from the equilibrium with price and quality discrimination and the optimal fee f^{p }if θ >0.(endnote e)
4 The optimal fee under balance billing
Glazer and McGuire are not explicit about how low the secondbest optimal fee f^{p }that maximizes the welfare function (11) is if balancebilling is allowed. Notably, they do not raise the question whether Medicare should pay any positive fee and, if so, whether the fee should be such that there are any feeonly patients. In the following, we therefore investigate how social surplus changes if Medicare reduces the fee or completely withdraws from the physician market.
In the preceding section, the number of feeonly patients of physician i has been shown to be
where v(s)  s ≤ 0 is the net social gain from quality, which is decreasing in s for s >0 because of the concavity of v. It is useful to define the level of quality
Hence,
where the definition of
When characterizing the optimal fee under balance billing, Glazer and McGuire restrict
themselves to the range between
What is the social surplus if the fee is reduced to
where s depends on f through equation (9). Social surplus is calculated by multiplying per capita net social value of quality with the number of feeonly patients in equilibrium and subtracting the distortion costs of the fee.
Proposition 1: If the payer is confined to the normal range of fee policy, i.e.
Proof: We have
Evidently, if θ is sufficiently high, then this difference is always negative, and
welfare is maximized by setting the lowest possible fee, which is
Being confined to f ≤ f*, we always deal with s ≥ 0. Hence, by (10), the first two factors at the lefthand side of (18) are both
negative. For (18) to be satisfied, 1/2 + v(s)  s >0 is therefore required, which implies
The intuition of Proposition 1 can be explained by considering the firstbest welfare
function
Obviously, welfare may be further increased by reducing the fee from
Proposition 2: If the payer can set any fee f ∈ [0;∞) and patients' willingness to pay is sufficiently high, then the firstbest welfare optimum W* = W^{p }= 0 can be implemented by setting f = 0 under balance billing.
Proof: For
If
This establishes that even in the secondbest world with θ >0, firstbest efficiency is implementable by the complete withdrawal of Medicare from the physician market. Provided that the willingness to pay is sufficiently high, all patients then become pricepayers and receive the optimal quality while there are no distortion costs so that the surplus function used by Glazer and McGuire becomes maximal. A further implication of Proposition 2 is that 'payer fee discrimination', an alternative fee policy proposed by Glazer and McGuire which discriminates the Medicare fee between feeonly and pricepaying patients, cannot further improve efficiency.(endnote j)
Figure 2 illustrates the shape of the welfare function (17). The graph of W^{p }(f) is constructed as the difference between the firstbest welfare W*(f) = (v(s(f))  s(f))(1 + v(s)  s(f))/2 and the distortion costs θf. W* has a corner maximum on the interval
Figure 2. SecondBest Welfare and Medicare payments f.
The central result obtained by Glazer and McGuire must therefore be strengthened:
moving from an equilibrium without balancebilling and optimal fee f^{o }to an equilibrium with balancebilling and optimal fee f = 0, not only a welfare improvement but the firstbest optimum can be attained. As
we show in the following section, however, this result depends crucially on the welfare
function (11) which only takes into account the efficiency of quality but does not
consider any distributional effects. From the patients' perspective, a Medicare fee
in the 'normal range'
5 Balance billing and patient welfare
An important aspect with respect to balance billing is how patients are affected by this policy. In particular, there is the concern that patients are made worse off if physicians are allowed to charge an additional price. In the models by Paringer, Mitchell and Cromwell as well as Zuckerman and Holahan which do not consider effects on quality, this can lead to the drastic effect that balance billing only raises the physician's rent [24]. An open question is the effect on patient welfare within the model by Glazer and McGuire. Is the positive effect of balance billing on quality dominant or are the quality gains transformed into higher rents for physicians? In this section, we take the patients' point of view and try to answer this question. Before we ask in Section 5.2 whether balance billing should be allowed, we first determine in Section 5.1 the optimal Medicare fee if balance billing is allowed. Finally, we consider the effects of 'fee discrimination', a policy proposed by Glazer and McGuire, on patient welfare in Section 5.3.
5.1 Should Medicare set a positive fee under balance billing?
To assess the effects of Medicare's policy on patient welfare, we need to specify
in more detail how Medicare's expenditures are financed. In the following, we assume
that the government collects a uniform contribution (1 + θ)f from each individual where f is the fee paid to physicians and θf are the administrative costs of Medicare per capita. Hence, the utility for feeonly
patients is given by
If Medicare sets a fee f above
Thus, patients who are treated at the fee only face a lower quality reduction than the price charged from pricepaying patients. This is illustrated in Figure 3 which is based on the cost savings function
Figure 3. Utility distribution for f = 0 and
where we set a = 1. It shows the utility distribution for f = 0 and for a fee
We measure patient welfare by the sum of utility of all patients.(endnote m) As we show in Appendix A.1, patient welfare under balance billing PW^{p }then corresponds to
Increasing Medicare's fee therefore lowers patient welfare if
where
If
By equations (7) and (19) we obtain for the price for balancebilled patients
This yields h(f, θ) = 0 for θ = 0 and
In Appendix A.2 we show that h(f) is increasing in f at
this result continues to hold as long as θ is below a critical value
Thus, for small values of θ even pricepaying patients are better off if f is raised above
Figure 4. Utility distribution for f = 0 and
We summarize our results in
Proposition 3: Under balancebilling setting,
Table 1 shows the results for a numerical simulation with the cost savings function (21)
for a = 1. Patient welfare if the Medicare fee is zero is given by PW^{p}(f = 0). For different values of θ, the Medicare fee
Table 1. Patient welfare under balance billing
• for θ = 0% or 10%, we have
• for θ = 0% to 20%, we have
Our result is in stark contrast to the social surplus analysis in section 4 where
f = 0 is the optimal fee level. In particular, setting the Medicare fee above
i.e., the decrease in physicians's profits is larger than the increase in patient welfare. For our numerical simulation with θ = 10%, Figure 5 shows how patient welfare increases even though social surplus falls. Patients are better off even though quality provision is less efficient on average.
Figure 5. Patient welfare and profits as a function of f, θ = 10%.
This result can be explained by the effects of Medicare's policy on physician competition. By raising f, competition for patients gets more intense and profits of physicians fall by more than the decrease in average quality. Especially patients in the middle get a better deal as physicians are willing to treat them at the feeonly. Although they receive lower quality, they get a more favorable offer than pricepaying patients as p > s. In addition, pricepaying patients may also be better off if Medicare's administrative cost markup θ is sufficiently small.
5.2 Should balance billing be allowed?
One of the central findings of Glazer and McGuire is that social surplus is generally higher if balance billing is allowed. However, balance billing also gives physicians the opportunity to increase their profits. It is therefore unclear whether patients also benefit if balance billing is permitted. We investigate this issue by taking a regime without balance billing and f^{o}, the corresponding optimal fee from an efficiency perspective, as a reference point (see equation (13)). The corresponding quality level s^{o}(f^{o}) is defined by equation (12), leading to patient welfare
Turning to a comparable regime with balance billing, we define
First, we assume that
since s^{o } v(s^{o}) ≥ 0 is implied by v(0) = 0,v'(0) = 1, v'(s) >0 and v″(s) <0. Thus, under balance billing the same quality for feeonly patients can be provided with a lower Medicare fee.
For patient welfare under balance billing (see equation (22)), we obtain
Using (25) yields for the difference of total utilities under the two regimes
The first term is negative since s > p for
Next, we turn to the case
Again if θ is sufficiently large, then average utility increases when balance billing is allowed.(endnote o) We can therefore conclude in
Proposition 4: For a given fee level f^{o }without balance billing, patient welfare can be increased by allowing balance billing if Medicare's administrative cost markup θ is sufficiently large.
Proposition 4 shows the main drawback of prohibiting balance billing. Inducing quality without balance billing is very costly if Medicare's administrative costs are high. Permitting balance billing allows to induce the same quality at a lower Medicare fee. The corresponding savings in administrative costs can exceed higher payments to physicians from patients who are balance billed.
In assessing Proposition 4, however, one has to keep in mind that neither f^{o }nor
Numerical simulations based the cost savings function (21) indicate that Proposition
4 can also be extended to optimally chosen fee levels. An example is shown in Table
2. For different values of θ, s^{o}* and f^{o}* are the optimal values without balance billing. Maximized patient welfare without
balance billing is denoted by PW^{o}(f^{o}*). Besides
Table 2. Prohibiting vs.allowing balance billing
It is also possible that allowing balance billing yields higher patient welfare already for θ = 0. The shape of the cost savings function v(s) is crucial for this result. This is shown in Figure 6. Setting a = 2 in the cost savings function (21) yields the optimal values f^{o}* = 2.00, f^{p* }= 1.92, PW^{o}(f^{o}*) = 7.201 and PW^{p}(f^{p}*) = 7.183 which implies that prohibiting balance billing is superior from the patients' perspective (see Figure 6(b)). For a = 1, however, we obtain f ^{o}* = 2.00, f^{p* }= 1.86, PW^{o}(f^{o}*) = 7.057 and PW^{p}(f^{p}*) = 7.126. Figure 6(a) shows that allowing balance billing is superior for all values of f if θ = 0.
Figure 6. Patient welfare as a function of f, θ = 0.
In sum, it depends on Medicare's administrative costs and the properties of the cost function whether allowing balance billing raises patient welfare. In contrast to Glazer and McGuire, we do not find that allowing balance billing is generally superior. In their analysis, only quality effects matter and allowing balance billing is better because it can always induce the same quality at a lower cost. From the patients' perspective, it also has to be taken into account that physicians charge a price from selected patients. This reduces patient welfare. As long as θ is small and inducing quality without balance billing is therefore not too costly, this profit effect may dominate and prohibiting balance billing leads to higher patient welfare. For example, if a = 2 and θ = 0, aggregate profits are Π^{o}(f^{o}*) = 0.333 and Π^{p}(f ^{p}*) = 0.513. But higher profits do not necessarily imply lower patient welfare. For a = 1 and θ = 0, the corresponding values are Π^{o}(f^{o}*) = 0.5 and Π^{p}(f^{p}*) = 0.574. Nevertheless, patient welfare is higher as the Medicare fee is lower (1.86 vs. 2.00) and the quality reduction for feeonly patients is significantly smaller (0.52 vs. 0.69) under balance billing.
5.3 Fee discrimination
Glazer and McGuire also analyze a regime of balance billing under which Medicare discriminates the fee depending on whether the physician treats patients at the fee only or balance bills them. Under this policy, physicians are reimbursed f + d if they do not charge their patients price p and f  d if they do, with d >0. Glazer and McGuire argue that such a fee policy is welfare improving based on the efficiency criterion if the extent of discrimination as measured by d is small and if the fee is set close to its optimum.(endnote p) However, as already mentioned in Section 4, Proposition 2 implies that the firstbest welfare level can be implemented by Medicare withdrawing form the market, ruling out any strictly positive efficiency effect of fee discrimination. The question remains to be answered whether fee discrimination can be justified if Medicare is assumed to be concerned about patient welfare.
We first derive the equilibrium price under fee discrimination. Since the payer increases the fee by d >0 for the treatment of feeonly patients and reduces it by the same amount in the other cases, physician i's profit becomes
with the firstorder conditions for a Nashequilibrium
Assuming symmetry, (28) can be solved for the equilibrium price under fee discrimination
Thus, two effects result from the introduction of fee discrimination by d on the price physicians charge. First, a direct effect implies that physicians increase the price just by d to compensate for the lower fee that they receive for the treatment of pricepaying patients. Note that this is a difference to variations in f where no such direct effect on the price exists. Second, an indirect effect on p works through the influence of d on the equilibrium level of quality. By substituting (30) into (29) for the symmetric case and differentiating, we can determine this effect of d on quality as follows
Hence, (31) is similar to ds/df in (10) with one difference. In contrast to (10), the sign of (31) now depends on the volume of the marginal cost savings from reduced quality v'(s). This results from the direct effect of d on the price found in (30). If marginal cost savings are high (v'(s) >1 ⇔ s <0), then quality is decreased in response to a marginal increase in d and the price is increased more than proportionally. This can be seen from differentiating (30)
Otherwise, due to (10) and (31), physicians react to an increase in the amount of fee discrimination by an enhanced quality and a less than compensating price increase. In this case, a marginal increase in d is similar to an increase of the fee f which induces physicians to offer a better quality and decrease the price.
Given this information, we are able to compute the total number of feeonly patients
and of pricepaying patients in the market with fee discrimination
In the absence of fee discrimination (d = 0), we have
which is unambiguously negative. Intuitively, fee discrimination with d >0 makes feeonly patients more attractive to physicians. Therefore, their number rises in equilibrium.
These expressions allow us to compare patient welfare with fee discrimination (FD) and without fee discrimination (NFD) under balance billing. Assuming
Clearly, this is zero for d = 0 and differentiation yields after some rearrangements
Based on this equation, we prove the following in Appendix A.5.
Proposition 5: A small amount of payer fee discrimination increases patient welfare if either
• θ is sufficiently high, or
• the fee is chosen so as to maximize patient welfare and this results in s >0.
Under these circumstances, Medicare paying a higher fee to physicians who renounce on balance billing can indeed be justified from the perspective of the patients. However, it cannot be excluded that fee discrimination lowers patient welfare. This may occur whenever the price increase for the price payers dominates the quality increase for the feeonly patients or if the fee is set so that s <0, which implies that fee discrimination actually lowers quality.
6 Conclusions
This paper has revisited the economics of 'balance billing' in the framework by Glazer and McGuire [1]. We analyzed the optimal Medicare policy from the perspective of patients and showed that a positive Medicare fee and a mixed system with pricepaying and feeonly patients can increase patient welfare under balance billing if the administrative costs of Medicare are sufficiently low. The intuition for this result is that a positive Medicare fee increases competition of physicians which lowers the total payment to physicians by Medicare and patients.
Furthermore, we examined the case for permitting balance billing. We showed that it depends on Medicare's administrative costs and the properties of the physicians' cost function whether allowing balance billing raises patient welfare. In contrast to Glazer and McGuire, we do not find that allowing balance billing is generally superior as balance billing allows physicians to increase their rents. However, both physicians' rents and patient welfare can be higher if balance billing is permitted. This is the case for sufficiently high administrative costs of Medicare. For some cost functions, patient welfare can be higher under balance billing even in the absence of administrative costs.
Finally, we considered the effects on patient welfare if Medicare discriminates the fee depending on whether the physician treats patients at the fee only or balance bills them. This policy can also help to raise patient welfare. This is the case if Medicare's administrative costs are high or if Medicare's optimal fee under balance billing implies lower quality for feeonly patients.
Our study relied on a model based on profitmaximizing physicians. It may be interesting to relax this assumption in future research. Altruistic physicians may be less inclined to provide lower quality to feeonly patients. Furthermore, we assumed symmetric information about the quality of physicians' services. To the extent that patients cannot judge the quality of services, the efficiency of balance billing may be questionable. Balancebilled patients may only receive nonmedical amenities such as shorter waiting times for nonurgent treatments. An interesting extension is also to allow patients to differ in ability to pay.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
MK and FS have carried out the analysis and written the paper together. Both authors read and approved the final manuscript.
Appendix
A.1 Patient welfare under balance billing
For
as p = 1 + c  f in this case. If
Using
Thus, patient welfare is
A.2 Properties of the function h(f)
In the following, we show that the function
is increasing in f at
At
and therefore
Since
as the function g(a) = (1 + a)/(3a  a^{2}) has the following properties: g(1) = 1 and g'(a) = (a^{2 }+ 2a  3)(3a  a^{2})^{2 }<0 for 0 ≤ a <1. Thus, g(a) >1 for 0 ≤ a <1 and
If θ = 0, increasing f beyond
this result continues to hold as long as θ is below a critical value
A.3 Physicians' profits under balance billing
Aggregate profits are given by (cf. (4))
For
Inserting p = (1 + s + v(s))/2 from (7) yields
Thus, we can summarize
For
At
Patient welfare for
and therefore
Equation (A.1) implies
Furthermore,
since
A.4 Patient welfare under fee discrimination
As we have shown in Appendix A.1, patient welfare under balance billing without fee discrimination (NFD) is
for
from (1), this can be rewritten as
Analogously, patient welfare under balance billing with fee discrimination (FD) is given by
Hence, the change in patient welfare ΔPW = PW^{FD } PW^{NFD }induced by fee discrimination is given by (33).
A.5 Proof of Proposition 5
The last term of equation
θ(s  v(s)) measures the effect of d on the distortion costs. This term is unambiguously positive for s ≠ 0 because s  v(s) >0 and increases with θ. Noting that the first two terms on the right hand side of (34) do not depend on θ, we can therefore conclude that a small amount of payer fee discrimination increases patient welfare if θ is sufficiently large, irrespective of the level of f chosen originally.
The first two terms in (34) account for two further effects of fee discrimination on patient welfare, namely the induced change in price and quality. As was shown above, if s <0, then a marginal increase in d leads to a higher price and a lower quality, which both lowers patient welfare. The sign of the first two terms in (34) is therefore negative in this case and counteracts the positive effect from the reduced distortion. By contrast, if s >0, we have ds/dd <0 by (31), meaning that the feeonly patients receive a higher quality. This has a positive impact on patient welfare. For s >0, the overall sign of the first two terms in (34) is therefore ambiguous in general since the price and quality effects work in opposite directions. However, we can show that the positive quality effect dominates if the fee is chosen so as to maximize patient welfare. Assuming an interior optimum, the first order condition is
In addition, (31) implies
Substituting into (A.2) yields
or,
This confirms that if the fee is chosen optimally and s >0, then even the first two terms in (34) are positive in sum and hence a small amount of payer fee discrimination increases patient welfare irrespective of the size of θ.
Endotes
Endnote a. McKnight provides a detailed history of the legislation on balance billing [7].
Endnote b. By assumption, there is no further price discrimination among pricepaying patients.
Endnote c. For quality competition to be effective, the righthand side of the equation must
always be larger or equal than zero, hence we need
Endnote d. This is a corrected version of the corresponding equation (9) in [1].
Endnote e. See p. 251 in [1].
Endnote f. See Section 2.4 in [1].
Endnote g. To derive the demand functions, we already needed the assumption that exceeds a certain minimum value, which must be strengthened at this point.
Endnote h. Note that
Endnote i. Each physician makes a profit of π^{i }= 1/2.
Endnote j. See Section 5.3 for a further evaluation of fee discrimination.
Endnote k. By inspection of the first derivative ∂W*/∂f = (v'(s)  1)(1/2 + v(s)  s)ds/d f there exists a local minimum for
Endnote l. As mentioned in the proof of Proposition 1, this local maximum of the secondbest welfare function (as well as the minimum) may not exist if θ is very high.
Endnote m. An alternative measure is the utility of the worstoff patient. It can be shown
that under balance billing with
Endnote n. For the critical value
Endnote o. Note that in this case all patients must be better off if patient welfare is higher
under balance billing: Under balance billing all patients pay price p = 1 + c for quality s = 0 and therefore obtain utility
Endnote p. See the proposition on page 252 in [1]. Their claim is that efficiency rises if d is small and f close to the secondbest level f ^{p }in their framework. However, in the proof, they refer to a situation where s approaches zero, which is associated with f = f*.
Acknowledgements
We thank Kurt Brekke, Friedrich Breyer, Oddvar Kaarboe, Erik Schokkaert and Astrid Selder for helpful comments and suggestions.
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