Patient selection in a mixed oligopoly market for health care: the role of the soft budget constraint

Abstract

Competition in hospital care is often implemented through mixed markets where public and private hospitals compete for patients. The optimality of this market form has long been debated in the literature. In this paper, we investigate the role of soft budget constraint in affecting patient selection within a mixed market. Patient selection is the undesired effect of hospital competition when three conditions are met: asymmetry in hospitals’ objectives, presence of hospital’s private information and inability to enforce hard budget constraint. The paper shows that soft budget is a pre-condition for the existence of patient selection. Our paper adds an important dimension to the existing literature which considers asymmetry of information as the only cause for this market failure. The understanding of the mechanisms leading to patient selection makes it possible for the regulator to design measures to reduce such undesirable effect.

Appendix

1.1 Derivation of equation (4)

Patients are indifferent between hospital A and hospital B when

$$ \alpha q_{{H_{A} }}^{i} - \gamma d = \alpha q_{{H_{B} }}^{i} - \gamma (1 - d)\quad (i = L,H) $$

Solving for d, we get

$$ d = \frac{\alpha }{2\gamma }\left( {q_{{H_{A} }}^{i} - q_{{H_{B} }}^{i} } \right) + \frac{1}{2}\quad i = L,H $$

(12)

where d represents the location of the marginal consumer. Substituting (2) in (12), we obtain

$$ d^{L} = \frac{\alpha }{2\gamma }[w(\hat{q}_{{H_{A} }} - \hat{q}_{{H_{B} }} ) + (1 - w)(\tilde{q}_{{H_{A} }} - \tilde{q}_{{H_{B} }} )] + \frac{1}{2} $$

$$ d^{H} = \frac{\alpha }{2\gamma }[(1 - w)(\hat{q}_{{H_{A} }} - \hat{q}_{{H_{B} }} ) + w(\tilde{q}_{{H_{A} }} - \tilde{q}_{{H_{B} }} )] + \frac{1}{2} $$

The demand for hospital j is obtained by multiplying the distance times the density which, given the unit length of the line, is equal to the share p for low and to (1 − p) for high-severity patients:

The demand can be written as

$$ D_{j}^{L} = \left\{ {\frac{\alpha }{2\gamma }[w(\hat{q}_{j} - \hat{q}_{h} ) + (1 - w)(\tilde{q}_{j} - \tilde{q}_{h} )] + \frac{1}{2}} \right\}p $$

$$ D_{j}^{H} = \left\{ {\frac{\alpha }{2\gamma }[(1 - w)(\hat{q}_{j} - \hat{q}_{h} ) + w(\tilde{q}_{j} - \tilde{q}_{h} )] + \frac{1}{2}} \right\}(1 - p) $$

where h ≠ j; h, j = H _A , H _B

1.2 Derivation of equation (9)

The budget constraint is met when

$$ [q^{*} - \tilde{q}_{{H_{A} }} - \hat{q}_{{H_{A} }} + E(\beta ) - \beta_{L} ]D_{{_{{H_{A} }} }}^{L} + [q^{*} - \tilde{q}_{{H_{A} }} - \hat{q}_{{H_{A} }} + E(\beta ) - \beta_{H} ]D_{{_{{H_{A} }} }}^{H} \ge 0 $$

(13)

Since all the reimbursement q* is turned into quality by the public hospital, that is, $\hat{q}_{{H_{A} }} + \tilde{q}_{{H_{A} }} = q^{*}$ and E(β) = pβ _L  + (1 − p)β _H, (13) can be written as

$$ (\beta_{H} - \beta_{L} )p(1 - p)d_{{_{{H_{A} }} }}^{L} + (\beta_{H} - \beta_{L} )p(1 - p)d_{{_{{H_{A} }} }}^{H} \ge 0 $$

Assuming the zero profit condition¹⁷ for the public hospital,

$$ \frac{\alpha }{2\gamma }[w(\hat{q}_{{H_{A} }} - \hat{q}_{{H_{B} }} ) + (1 - w)(\tilde{q}_{{H_{A} }} - \tilde{q}_{{H_{B} }} )] + \frac{1}{2} = \frac{\alpha }{2\gamma }[(1 - w)(\hat{q}_{{H_{A} }} - \hat{q}_{{H_{B} }} ) + w(\tilde{q}_{{H_{A} }} - \tilde{q}_{{H_{B} }} )] + \frac{1}{2} $$

using (8), we get

$$ 2w[(1 - \phi )q^{*} - \hat{q}_{{H_{B} }} ] - [(1 - \phi )q^{*} - \hat{q}_{{H_{B} }} ] + [\phi q^{*} - \tilde{q}_{{H_{B} }} ] - 2w[\phi q^{*} - \tilde{q}_{{H_{B} }} ] = 0 $$

and finally the condition for φ

$$ \phi = \frac{1}{2} + \frac{{\tilde{q}_{{H_{B} }} - \hat{q}_{{H_{B} }} }}{{2q^{*} }} $$

1.3 Derivation of equation (10)

In the market, there are p low-severity patients and (1 − p) high-severity patients. The former prefer (by the weight w > 1/2) hotel-related quality ($\hat{q}$), whereas the latter prefer health-related quality ($\tilde{q}$).

Weighting the number of each severity type patients [p and (1 − p)] by their preferences with respect to quality, the following equation can be set:

$$ p[w\hat{q} + (1 - w)\tilde{q}] + (1 - p)[(1 - w)\hat{q} + w\tilde{q}] $$

or equivalently:

$$ \hat{q}(1 - \delta ) + \tilde{q}\delta $$

where

$$ \delta = p + w - 2pw $$

Assuming the reimbursement equal to q*, and $\hat{q} + \tilde{q} = q*$, the only way to allocate the reimbursement in order to respect the patient preferences is:

$$ \begin{aligned} \hat{q}_{{H_{A} }} & = (1 - \delta )q^{*} \\ \tilde{q}_{{H_{A} }} & = \delta q^{*} \\ \end{aligned} $$

that is, $\varphi$ has to be set equal to $\delta$.

1.4 Proof of proposition 1

By definition, we observe patient selection when

$$ \frac{{D_{{_{{H_{B} }} }}^{L} }}{{D_{{_{{H_{B} }} }}^{H} }} \ne \frac{p}{(1 - p)} $$

where

$$ D_{{_{{H_{B} }} }}^{L} = \left\{ {\frac{\alpha }{2\gamma }[w(\hat{q}_{{H_{B} }} - \hat{q}_{{H_{A} }} ) + (1 - w)(\tilde{q}_{{H_{B} }} - \tilde{q}_{{H_{A} }} )] + \frac{1}{2}} \right\}p $$

$$ D_{{_{{H_{B} }} }}^{H} = \left\{ {\frac{\alpha }{2\gamma }[(1 - w)(\hat{q}_{{H_{B} }} - \hat{q}_{{H_{A} }} ) + w(\tilde{q}_{{H_{B} }} - \tilde{q}_{{H_{A} }} )] + \frac{1}{2}} \right\}(1 - p) $$

and therefore, the necessary and sufficient condition for patient selection can be written as:

$$ \frac{{[w(\hat{q}_{{H_{B} }} - \hat{q}_{{H_{A} }} ) + (1 - w)(\tilde{q}_{{H_{B} }} - \tilde{q}_{{H_{A} }} )]}}{{[(1 - w)(\hat{q}_{{H_{B} }} - \hat{q}_{{H_{A} }} ) + w(\tilde{q}_{{H_{B} }} - \tilde{q}_{{H_{A} }} )]}} \ne 1 $$

(14)

Substituting (9) in (8): $\hat{q}_{{H_{A} }} = \frac{{q^{*} }}{2} + \frac{{\hat{q}_{{H_{B} }} - \tilde{q}_{{H_{B} }} }}{2}$; $\tilde{q}_{{H_{A} }} = \frac{{q^{*} }}{2} + \frac{{\tilde{q}_{{H_{B} }} - \hat{q}_{{H_{B} }} }}{2}$

The LHS of 14 is equal to 1; it can be immediately verified as follows:

$$ \frac{{w\left( {\frac{{\hat{q}_{{H_{B} }} + \tilde{q}_{{H_{B} }} }}{2} - \frac{{q^{*} }}{2}} \right) + (1 - w)\left( {\frac{{\hat{q}_{{H_{B} }} + \tilde{q}_{{H_{B} }} }}{2} - \frac{{q^{*} }}{2}} \right)}}{{w\left( {\frac{{\hat{q}_{{H_{B} }} + \tilde{q}_{{H_{B} }} }}{2} - \frac{{q^{*} }}{2}} \right) + (1 - w)\left( {\frac{{\hat{q}_{{H_{B} }} + \tilde{q}_{{H_{B} }} }}{2} - \frac{{q^{*} }}{2}} \right)}} = 1 $$

Therefore, we can conclude that $\frac{{D_{{_{{H_{B} }} }}^{L} }}{{D_{{_{{H_{B} }} }}^{H} }} = \frac{p}{(1 - p)}$ or in other words that the condition for patient selection is never met (no patient selection will occur).

1.5 Proof of proposition 2

To prove that R ^EXC > R ⁰ is necessary to show that $\frac{{D_{{H_{B} }}^{L} }}{{D_{{H_{B} }}^{H} }} > R^{0}$ or equivalently that $\frac{{d_{{_{{H_{B} }} }}^{L} }}{{d_{{_{{H_{B} }} }}^{H} }} > 1.$

This result clearly emerges from the equation below (where the quality levels have been substituted in the demand functions) which is always greater than 1 (remember that w > 1/2, and by definition, $q^{*} \ge \hat{q}_{{H_{B} }}$): $\frac{{w\hat{q}_{{H_{B} }} - (1 - w)q^{*} }}{{(1 - w)\hat{q}_{{H_{B} }} + wq^{*} }} > 1$. Hence, we can conclude that the condition R ^EXC > R ⁰ is always true.

1.6 Proof of proposition 3

Since the condition for patient selection, in the excellence and equity cases, requires the following conditions to be met (Table 3):

Table 3 Excellence and equity objective

From the previous proof (proof of proposition 2), we know that the condition R ^EXC > R ⁰ is always met. However, we can state that the scope for cream skimming depends on the ratio $\frac{\gamma }{\alpha }$ (in the limiting cases for $\gamma \to \infty ;\alpha \to 0$, the strategic space for cream skimming is set apart).

From Table A1, it emerges clearly that when R ^EQ > R ⁰, then also the condition R ^EXC > R ⁰ has to be met, provided δ = p + w − 2pw < 1, which in turn implies that R ^EXC > R ^EQ > R ⁰, ∀p ∈ (0, 1).

Therefore, assuming condition to be met, then, although the value of p does not affect the sign R ^EXC–R ^EQ, it determines, jointly with w, the intensity of cream skimming.