Optimal Reimbursement Policy in Health Care: Competition, Ownership Structure and Quality Provision

Rune Stenbacka; Mihkel Tombak

doi:10.1515/bejeap-2017-0122

Published by De Gruyter January 23, 2018

Optimal Reimbursement Policy in Health Care: Competition, Ownership Structure and Quality Provision

Rune Stenbacka and Mihkel Tombak

From the journal The B.E. Journal of Economic Analysis & Policy

https://doi.org/10.1515/bejeap-2017-0122

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Abstract

We analytically characterize the effects of ownership and competition in the healthcare industry on quality provision, market coverage and optimal reimbursement policy. A for-profit monopoly selects a lower quality than a nonprofit supplier, and the socially optimal reimbursement rate with a nonprofit monopoly exceeds that with a for-profit monopoly. We establish that the optimal repayment policy is invariant to the introduction of competition by a for-profit high-quality supplier. Thus, market coverage is invariant to the introduction of competition, whereas consumers with a higher willingness to pay for quality are better off with competition.

Keywords: for-profit vs nonprofit healthcare provision; competition in healthcare; healthcare quality; quality differentiation; mixed duopoly

JEL Classification: I11; I18; L10

Appendix

A The quality equilibrium with entry by a for-profit high-quality supplier

The objective function in (7) is strictly increasing in q1NP for λ^1−λ~1. In order to see this we observe that Γ′(q1NP)=N2λ¯[(λ^1+λ~1)(λ^1−λ~1)+q1NP(∂λ^1∂q1NP−∂λ~1∂q1NP]\gt;0, because ∂λ^1∂q1NP\gt;0 and ∂λ~1∂q1NP\lt;0. Thus, faced with entry from a for-profit supplier the nonprofit organization selects the quality given by

(19)Nλ¯∫λ~1λ^1(R−cNPq1NP)dλ=Nλ¯(R−cNPq1NP)(λ^1−λ~1)=0.

Clearly, in light of (19), the standard quality q1NP=RcNPimplies that NP indeed meets the zero profit constraint. Thus, the quality provision by the nonprofit is invariant to for-profit entry with premium quality. In order to make sure that q1NP=RcNPis part of a uniquely determined equilibrium where NP is active we still have to check the proposed equilibrium qualities do not violate the constraint λ^1\gt;λ~1.

We next shift our attention to the decision of the profit-maximizing supplier of the premium quality. By application of Leibniz’ integral rule with variable limits we find that the first-order condition associated with the optimization problem (9) is given by

(20)∂π1FP∂q1FP=Nλ¯[−cFP∫λ^1λ¯⁡dλ−(R+p−cFPq1FP)∂λ^1∂q1FP]=0,

where ∂λ^1∂q1FP=−λ^1(q1FP−q1NP). Substituting q1NP=RcNP and solving the first-order condition (20) demonstrates that the optimal premium quality for the private supplier is given by the solution to the quadratic equation

cFPλ¯(q1FP)2−2cFPλ¯q1NPq1FP+cFPλ¯(q1NP)2−(p2+pR(1−cFPcNP))=0.

Solution of this quadratic equation yields the optimal premium quality

q1FP=RcNP+p(p+R(1−cFPcNP))cFPλ¯.

It can directly be verified that the objective function in (9) is strictly concave, meaning that the solution to (20) yields the equilibrium quality.

It remains to verify that the equilibrium configuration q1NP=RcNPand q1FP=RcNP+p(p+R(1−cFPcNP))cFPλ¯is consistent with the constraint λ^1\gt;λ~1. The inequality λ^1\gt;λ~1 is equivalent to p\gt;ps(cNP)2λ¯cFP[p+R(1−cFPcNP)]p. A sufficient condition for this to hold is that the expression under the square root is smaller than or equal to 1, which is equivalent to Rp(1−cFPcNP)⩽λ¯cFP(cNP)2−1.

QED

B The quality equilibrium with entry by a for-profit low-quality supplier

The objective function in (17) is strictly increasing in q2NP, because Γ′(q2NP)=Nq2NP2λ¯[λ¯2+pq2NP+q2FP(q2NP−q2FP)3]\gt;0. Thus, faced with entry from a for-profit supplier the nonprofit organization selects the quality given by

(21)Nλ¯∫λ^2λ¯2⁡(R+p−cq2NP)dλ=Nλ¯(R+p−cq2NP)(λ¯−λ^2)=0.

Clearly, in light of (21), the premium quality q2NP=R+pcimplies that NP indeed meets the zero profit constraint. In order to make sure that q2NP=R+pcis part of a uniquely determined equilibrium where NP is active we still have to check the proposed equilibrium qualities do not violate the constraint λ^2\gt;λ~2.

We next shift our attention to the decision of the profit-maximizing supplier of the lower quality. The objective function (16) is equivalent to

maxq2FPNλ¯(R−cq2FP)(λ^2−λ~2),

which after rearrangement yields the first-order condition,

(R−cq2FP)[p(q2NP−q2FP)2+Rs(q2FP)2]=c[p(q2NP−q2FP)−Rsq2FP].

Simplification shows that this is a quadratic equation in q2FP with the solution

q2FP=R+pcRs(p+Rs),q2NP=Rs(p+Rs).

The second-order derivative of the objective function for FP is given by

2Nλ¯(p(q2NP−q2FP)2(R−cq2NP)(q2NP−q2FP)−sR2(q2FP)3).

For q2NP=R+pc we can conclude that the second-order derivative of FP’s profit function is

2Nλ¯(−p2(q2NP−q2FP)3−sR2(q2FP)3)\lt;0,

meaning that the first-order condition characterizes the equilibrium.

It remains to verify that the equilibrium configuration q2FP=R+pcRs(p+Rs) and q2NP=R+pc is consistent with the constraint λ^2\gt;λ~2. This constraint is satisfied because λ^2−λ~2=(p+Rs)cR+p[1−s]\gt;0.

QED

Acknowledgements

The authors thank Alberto Galasso, Nico Lacetera, Peter Landry, Topi Miettinen, Staffan Ringbom, Gabor Virag, participants at the MACCI2014-conference at Mannheim University and ZEW as well as two anonymous referees for valuable comments. Financial support from the Hanken School of Economics Foundation is gratefully acknowledged.

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Published Online: 2018-1-23

Optimal Reimbursement Policy in Health Care: Competition, Ownership Structure and Quality Provision

Abstract

Appendix

A The quality equilibrium with entry by a for-profit high-quality supplier

B The quality equilibrium with entry by a for-profit low-quality supplier

Acknowledgements

References

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