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Differential Games in Health-Care Markets: Models of Quality Competition with Fixed Prices

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Advances in Dynamic and Mean Field Games (ISDG 2016)

Part of the book series: Annals of the International Society of Dynamic Games ((AISDG,volume 15))

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Abstract

We present a review of models that investigate quality competition in health-care markets under price regulation, taking a differential game approach. We analyse and discuss three different variants of a unified modelling framework, each of which incorporates specific institutional and behavioural characteristics of health-care markets. In each case we derive and compare equilibrium strategies under open-loop and feedback closed-loop information structures. We also address potential policy implications from these analyses.

This Chapter represents the reference of the plenary talk of R. Cellini at the 17th ISDG (Urbino, July, 2016); the content is based on Brekke et al. [3, 4] and Siciliani et al. [46]. We thank the participants at the ISDG, and the colleagues that provided us with comments and criticism on this work and the articles upon which it bases. In particular, we would like to mention T. Basar, G. Feichtinger, L. Grilli, F. Lamantia, G. Zaccour for helpful comments. The responsibility remains on the authors alone.

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Notes

  1. 1.

    Countries where price is not fixed for segments of the health-care market include the USA, Switzerland, France, and the Netherlands. Many issues need to be taken into account for the question of whether prices should be regulated or not in the health-care markets. One important issue is how endogenous pricing affects quality provision and whether competition along both dimensions (price and quality) will lead to a socially optimal quality provision or not. See Brekke, Siciliani, and Straume [10] for a discussion.

  2. 2.

    Even with linear transportation costs, a higher number of providers can lead to higher-quality provision if treatment costs are convex. In this case, a higher number of providers, which leads to lower demand for each provider, will increase the price-cost margin and therefore give each provider a stronger incentive to attract patients by raising quality.

  3. 3.

    See, e.g. Ma and Burgess [37], Wolinsky (1997) and Brekke, Nuscheler, and Straume [5].

  4. 4.

    The empirical evidence, however, is ambiguous: Kessler and McClellan [32] and Tay [47] find a positive effect, Gowrisankaran and Town [25] find a negative effect and Shen [44] finds mixed effects, while Shortell and Hughes [45] find no significant effects of competition on quality under regulated prices. Gaynor [22] offers a survey of theoretical and empirical literature on the relationship between competition and quality in the health sector. Recent evidence from England suggests that competition increases quality if prices are fixed ([15, 23]; Bloom et al. 2016).

  5. 5.

    The lower is γ, the more sluggish is demand: the parameter γ is therefore an inverse measure of the degree of demand sluggishness in the market. When time is discrete, it has to be assumed that 0 < γ < 1 to avoid that the adjustment is in excess to the disequilibrium; in continuous time, it is sufficient to assume γ > 0.

  6. 6.

    Rigid constraints, corresponding to assumptions, e.g. \(D\leq \overset {\_}{D}\) and \(\lim _{D\rightarrow \overset {\_}{D}}C(D,\ldots )=+\infty \) are rare to observe in the real world. Hospitals can expand capacity by, for instance, hiring more doctors or nurses or inducing existing staff to work overtime.

  7. 7.

    In reality, players may not have an infinite time horizon but reasonably long finite horizons. However, in general, the optimal path does not differ significantly from the solution with a very large but finite horizon, and the convenience of working with an infinite horizon model may be worth the loss of realism (see Léonard and van Long [35], p. 285).

  8. 8.

    For an introduction to the literature on motivated agents in the broader public sector, see, e.g. Francois [21] and Glazer [24]; for a specific focus on workers’ motivation in the health-care sector, see, inter alia, Ellis and McGuire [20], Eggleston [19] and Kaarbøe and Siciliani [30]. Brekke, Siciliani and Straume [9] offer an extensive discussion of the assumption of motivated providers, with further references to relevant literature, including experimental evidence.

  9. 9.

    In the USA, the State of New York was among the first to introduce ‘cards for hospital quality’ and for this reason has been intensively investigated in the empirical literature. There is evidence that market shares may be influenced by report cards with providers with better reports having larger market shares. Cutler et al. [16] show that hospitals with high mortality rates experience a 10% reduction in coronary bypasses, but this is not the case for hospitals with low mortality rates. Mukamel et al. [40] also find that higher mortality rates reduce market shares. Dranove and Sfekas [18] find that hospitals with bad reports have a smaller market share but only after accounting for the prior beliefs of the patients.

  10. 10.

    See Section 4.2. of Siciliani et al. [46] for a model under the open-loop solution.

  11. 11.

    We refer to Dockner et al. [17] for a general discussion about pros and cons of different solution concepts; in Brekke et al. [3, 4] and Siciliani et al. [46], there is a short discussion applied to the specific cases at hand.

  12. 12.

    This case arises, for example, when the marginal cost of quality and the marginal cost of provision are constant but quality and investment are complements, which implies \(C_{I_{i}q_{i}}<0\). The equilibrium is still a saddle point if \(\det (J)=-\delta (\delta +\rho )-(2\delta +\rho )C_{I_{i}q_{i}}/C_{I_{i}I_{i}}<0\). If \(\det (J)>0\), then we have an unstable node.

  13. 13.

    There are four possible solutions of which three are unstable. We focus on the stable solution.

  14. 14.

    However, alternative choices are possible as far as the measurement of the altruistic component is concerned, for instance, by considering the average or the marginal utility of patients instead of the aggregate surplus. Different modelling choices may affect the linear-quadratic structure of the problem.

  15. 15.

    In order to ensure that the second-order conditions are met, we assume that the parameters β and τ take values such that βτ > 1. In order to ensure that steady-state quality is always non-negative, we have to assume that the price is above a certain threshold level; specifically, we assume that \(p\geq \overline {p} :=c-\alpha \left ( v+\tau \left ( 1/2+\rho /\gamma \right ) \right ) \). A further parameter condition requiring α smaller than a threshold is sufficient to ensure stability of the open-loop solution (in the saddle sense) and the full stability of the feedback solution.

  16. 16.

    It is easy to show that \(\frac {\partial q_{i}}{\partial (1-D)}<0\) and \(\frac {\partial q_{j} }{\partial D}<0\) if \(\alpha <\frac {\beta \tau \left ( 2\rho +5\gamma \right ) }{2\gamma }\).

  17. 17.

    Note that provider motivation (α > 0) still ensures that quality levels are positive even if providers face negative price-cost margins. With purely profit-oriented providers, interior solutions would not exist if p < c.

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Author information

Authors and Affiliations

  1. Department of Economics, Norwegian School of Economics, Helleveien 30, N-5045, Bergen, Norway

    Kurt R. Brekke

  2. Department of Economics, University of Catania, Corso Italia 55, I95129, Catania, Italy

    Roberto Cellini

  3. Department of Economics and Related Studies, University of York, Heslington, York, YO10 5DD, UK

    Luigi Siciliani

  4. Department of Economics/NIPE, EEG, University of Minho, Braga, Portugal

    Odd Rune Straume

  5. Department of Economics, University of Bergen, Bergen, Norway

    Odd Rune Straume

Authors
  1. Kurt R. Brekke
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  2. Roberto Cellini
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  4. Odd Rune Straume
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Correspondence to Roberto Cellini .

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Editors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, Nova Scotia, Canada

    Joseph Apaloo

  2. Department of Mathematics, University of Padova, Padova, Italy

    Bruno Viscolani

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Brekke, K.R., Cellini, R., Siciliani, L., Straume, O.R. (2017). Differential Games in Health-Care Markets: Models of Quality Competition with Fixed Prices. In: Apaloo, J., Viscolani, B. (eds) Advances in Dynamic and Mean Field Games. ISDG 2016. Annals of the International Society of Dynamic Games, vol 15. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-70619-1_7

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