Individual decisions
We begin by introducing the model framework for individual quality decisions. For this, we consider a Salop model with an exogenously fixed number of three hospitals that compete in terms of treatment quality (Salop,
1979). A unit mass of patients is uniformly distributed on a circle. Patients receive medical treatment at hospitals that are equidistantly located along the circle. A patient’s utility depends on the quality level
\(q_{i}\) received in hospital
i with
\(i \in \left\{ {1,2,3} \right\}\), as well as on the travel distance between the hospital’s location
\(x_{i}\) and the patient’s location
\(z\). The disutility from traveling is measured by
\(t > 0\). Patients are fully insured, i.e., prices for treatment do not affect their utility. Furthermore, it is assumed that the ‘basic’ valuation of treatment
v is sufficiently large to ensure that receiving treatment is always preferred to remaining untreated.
8 Given the hospital’s location
\(x_{i}\) and the patient’s location
\(z\), the patient’s utility
\(u_{{z,x_{i} }} \) is given by:
$$ u_{{z,x_{i} }} = v + q_{i} - t \left| {z - x_{i} } \right| . $$
(1)
The total utility of patients being treated by the hospital i is
\(B_{i} = \mathop \smallint \limits_{o}^{{\hat{z}_{i}^{i + 1} }} \left( {v + q_{i} - ts} \right)ds + \mathop \smallint \limits_{o}^{{\hat{z}_{i} - 1}} \left( {v + q_{i} - ts} \right)ds, \) which determines the contribution to the patient in the following. It can be shown that hospital
i’s demand
\(D_{i}\) depends on the quality choices of all three hospitals active in the market and is given by:
$$ D_{i} = \frac{1}{3} + \frac{{2 q_{i} - \mathop \sum \nolimits_{j \ne i} q_{j} }}{2 t}. $$
(2)
Hospitals compete for patients in terms of quality.
9 Since prices
p for treatment are exogenously given by a regulator and marginal costs
\(c\) > 0 per quality are constant, hospital
i’s profit function can be written as:
$$ {\uppi }_{i} = \left( {p - c q_{i} } \right) D_{i} . $$
(3)
The three competing hospitals simultaneously choose their quality level in order to maximize their profit function as stated in Eq. (
3).
10 If we take the first-order condition (FOC), we can solve for the best-quality-response function for each hospital:
$$ q_{i} \left( {q_{j} } \right) = \frac{1}{2}\frac{p}{c} + \frac{{\mathop \sum \nolimits_{j \ne i} q_{j} }}{4} - \frac{t}{6} . $$
(4)
The corresponding symmetric Nash equilibrium quality
\(q_{i}^{*} \) and profit level
\({\uppi }_{i}^{*}\) of hospital
i is given by:
$$ q_{i}^{*} = \frac{p}{c} - \frac{t}{3} {\text{and}} {\uppi }_{i}^{*} = \frac{c t}{9} . $$
(5)
To derive theoretical benchmarks and our behavioural expectations for the strategic nature of quality provision within our experimental design, we now introduce our experimental parametrization of the formal model. We chose our parameters in a way that satisfies the participation constraint for patients—that is, demanding treatment even under the hospital’s minimal quality provision of 1 (
\(v = 5\)) and that patients do not travel beyond one of their neighbouring hospitals to receive a treatment (
\(t = 36\)). Regulated prices are set at
\(p = 44\) and treatment costs at
\(c = 2\). Subjects could choose a quality level
\(q_{i} \in \left\{ {1,2,3, \ldots ,13} \right\}\).
11
Based on this parametrization, we introduce some relevant benchmarks (see Table
2). The symmetric Nash equilibrium is reached at a joint quality level of 10, where each hospital makes a profit of 8 Taler
12 while the total utility of patients (contribution to patients) is 4 Taler. The patient optimum is at a quality level of 13, with a lower per hospital profit of 6 Taler and a higher contribution to the patient of 5 Taler. When all hospitals choose the minimum quality level of 1, the joint profit maximum (JPM) is reached. Every hospital gets a profit of 14 Taler in this period, and 1 Taler goes to the patient population. The benchmark defect refers to defecting from this collusive JPM. The optimal defection choice for purely profit maximizing providers is a quality level of 5 or 6,
13 both leading to profits of 15.11 Taler (a full profit and patient utility table can be found in Appendix A.1).
Table 2
Theoretical benchmarks
Nash equilibrium | 10 | 8 | 4 | 0 |
Patient optimum | 13 | 6 | 5 | -0.33 |
JPM | 1 | 14 | 1 | 1 |
Defect from JPM | 5 or 6 | 15.11 | 2.67 or 3.19 | |
Next, we introduce a measure for the degree of cooperation within a market. Following Suetens and Potters (
2007) and Potters and Suetens (
2009), we translate these quality benchmarks into degrees of cooperation. This gives us a straightforward indication about how collusive a healthcare market actually is. The degree of cooperation for healthcare market
\(k\) in period
\(t\) is calculated by:
$$ \rho_{kt} = \frac{{Avg. Market Quality_{kt} - Quality^{Nash} }}{{Quality^{JPM} - Quality^{Nash} }}. $$
(6)
Given our parametrization, it holds that \(\rho_{kt} = 0\) for average quality choices at the non-cooperative Nash equilibrium of 10. The JPM would result in \(\rho_{kt} = 1\), while a uniform quality choice at the patient optimum would result in \(\rho_{kt} = - 0.33\).
So far, our theoretical framework assumes pure profit-maximizing hospitals. However, when assuming semi-altruistic hospitals, similar to Brekke et al., (
2011,
2017), a hospital
\(i\)’s objection function can be written as:
$$ {\Omega }_{i} = \left( {p - c q_{i} } \right) \cdot D_{i} + \alpha \cdot B_{i} $$
(7)
where
\(B_{i} = \mathop \smallint \limits_{o}^{{\hat{z}_{i}^{i + 1} }} \left( {v + q_{i} - ts} \right)ds + \mathop \smallint \limits_{o}^{{\hat{z}_{i} - 1}} \left( {v + q_{i} - ts} \right)ds \) is the total utility of the patients being treated in hospital
\(i\) and
\(\alpha > 0\) is a measure for the degree of altruism.
14 The optimal Nash equilibrium quality, best response function, and strategic relationship thus depend on the degree of altruism (see Appendix A.3 for the formal model equations). The best response function allows hospitals’ quality levels to be strategic complements as well as a strategic substitutes. However, for semi-altruistic hospitals that maximize their objective functions, quality would be a strategic complement. Moreover, we show that our parametrization quality levels of semi-altruistic hospitals are higher (i.e.
\(q_{i\alpha }^{*} > q_{i}^{*} )\) and profits are lower compared to pure profit-maximizing hospitals (i.e.
\( {\uppi }_{i\alpha }^{*} < {\uppi }_{i}^{*} )\).
The focus of this paper is to better understand strategic behaviour within hospital markets. Therefore, we next translate our expectations to the degree of cooperation. Given our parametrization, the degree of cooperation is
\(\rho_{kt} = - 0.33\) for uniform quality choices at the patient utility optimum of 13 (see Table
2). Hence, in contrast to profit-maximizing providers, for which the Nash equilibrium predicts a symmetric quality choice of 10 that translates into a degree of cooperation of 0, markets with semi-altruistic providers should supply higher quality levels and have a negative degree of cooperation. In Potters and Suetens (
2009), negative values of the degree of cooperation
\(\rho_{kt}\) are interpreted as high competitiveness beyond the Nash equilibrium. In our setting, however, negative values also correspond to average quality choices towards the patient utility optimum.
15
Team decisions
Next, we derive hypotheses for team decisions based on the existing empirical and experimental evidence.
16 Comparing individual decisions with team decisions relates to the discussion about the ‘unitary player assumption’ and whether team behaviour and individual behaviour are equivalent. This might be even more important in healthcare markets, as providers’ preferences concerning profits and patient benefit might be heterogeneous in the internal decision-making process. While health economic experiments investigating healthcare provider behaviour have observed heterogeneity in the altruism of providers (Godager & Wiesen,
2013; Brosig-Koch et al.,
2016,
2017a), little is known about team decisions with potentially heterogeneously altruistic healthcare providers.
There is a vast literature on group behaviour in different economic experiments (see, for example, Engel,
2010 for a survey).
17 Some studies investigate group behaviour in market competition experiments with homogenous goods. Bornstein et al. (
2008), for instance, find that teams have a harder time establishing tacit collusion than individuals in Bertrand duopolies. Bornstein and Gneezy (
2002) also find that teams converge to the competitive solution quicker. Raab and Schipper (
2009), in contrast, do not find differences for three firm Cournot oligopolies. In our context of quality competition between groups, there is no evidence yet on how altruistic motives interact with the strategic nature of quality.
For team decisions, the voting rule—here, the majority voting rule—might also impact decisions. For example, Gillet et al. (
2011) showed that different decision rules—either by single persons who decide for their team members (‘CEOs’) or by a consensus or majority voting rule in a Bertrand pricing game—affect asking prices. In particular, for the comparable situation in which no cartel is formed, they find that prices are higher with a CEO and majority voting rule compared to the consensus rule and individual treatment. For profit-maximizing markets, we might expect lower quality levels and hence a higher degree of cooperation with the majority voting rule compared to individual decision making. However, there is no evidence for semi-altruistic teams applying the majority voting rule.