Optimal funding coverage in a mixed oligopoly with quality competition and price regulation

Abstract

We study the optimal design of a public funding scheme in a mixed oligopoly setting (with applications to health care and education) with one welfare-maximising public provider and two profit-maximising private providers, where all providers compete on quality and where providers included in the public funding scheme are subject to price regulation. We find that the first-best solution cannot be implemented without including (at least) one of the private providers in the public funding scheme. However, inclusion of only one of the private providers is sufficient to induce the first-best outcome. Such inclusion allows for the elimination of a negative competition externality between the private providers that, all else equal, yields underprovision of quality.

Appendix

1.1 A. Equilibrium existence

1.1.1 No private provider is included in the public funding scheme

In the pricing subgame, the second order conditions are satisfied,

$$\begin{aligned} \frac{\partial ^{2}\pi _{2}}{\partial p_{2}^{2}}= \frac{\partial ^{2}\pi _{3}}{\partial p_{3}^{2}}=-\frac{2}{t}<0, \end{aligned}$$

(A1)

and equilibrium stability requires that the Jacobian is negative definite, which is easily verified:

$$\begin{aligned} \frac{\partial ^{2}\pi _{2}}{\partial p_{2}^{2}}\frac{\partial ^{2}\pi _{3}}{ \partial p_{3}^{2}}-\frac{\partial ^{2}\pi _{2}}{ \partial p_{2}\partial p_{3}}\frac{\partial ^{2}\pi _{3}}{\partial p_{2}\partial p_{3}}=\frac{15}{4t^{2}} >0. \end{aligned}$$

(A2)

In the quality subgame, there are two sets of conditions that do not trivially hold. First, the problem of each profit maximising provider is well-behaved if

$$\begin{aligned} \frac{\partial ^{2}\pi _{2}}{\partial q_{2}^{2}}=\frac{\partial ^{2}\pi _{3}}{\partial q_{3}^{2}}=\frac{1}{225t}\left( 98\beta ^{2}-225kt\right) <0, \end{aligned}$$

(A3)

and the problem of the welfare-maxmising provider is well-behaved if

$$\begin{aligned} \frac{\partial ^{2}W}{\partial q_{1}^{2}}=\frac{1}{9t}\left( 8\beta ^{2}-9kt\right) <0. \end{aligned}$$

(A4)

Second, the Nash equilibrium is locally stable if the Jacobian is negative definite, which requires

$$\begin{aligned} \frac{\partial ^{2}W}{\partial q_{1}^{2}}\frac{\partial ^{2}\pi _{2}}{ \partial q_{2}^{2}}-\frac{\partial ^{2}\pi _{2}}{\partial q_{2}\partial q_{1} }\frac{\partial ^{2}W}{\partial q_{1}\partial q_{2}}=\frac{kt\left( 225kt-298\beta ^{2}\right) +56\beta ^{4}}{225t^{2}}>0 \end{aligned}$$

(A5)

and

$$\begin{aligned} \left| \begin{array}{ccc} \frac{\partial ^{2}W}{\partial q_{1}^{2}} &{} \frac{\partial ^{2}W}{\partial q_{1}\partial q_{2}} &{} \frac{\partial ^{2}W}{\partial q_{1}\partial q_{3}} \\ \frac{\partial ^{2}\pi _{2}}{\partial q_{2}\partial q_{1}} &{} \frac{\partial ^{2}\pi _{2}}{\partial q_{2}^{2}} &{} \frac{\partial ^{2}\pi _{2}}{\partial q_{2}\partial q_{3}} \\ \frac{\partial ^{2}\pi _{3}}{\partial q_{3}\partial q_{1}} &{} \frac{\partial ^{2}\pi _{3}}{\partial q_{3}\partial q_{2}} &{} \frac{\partial ^{2}\pi _{3}}{ \partial q_{3}^{2}} \end{array} \right| =-\frac{k\left( 5kt-6\beta ^{2}\right) \left( 25kt-14\beta ^{2}\right) }{125t^{2}}<0. \end{aligned}$$

(A6)

All the above conditions are satisfied if $k\ge {\underline{k}}$, where ${\underline{k}}$ is explicitly given by (12).

1.1.2 Inclusion of one private provider in the public funding scheme

In the quality subgame, there are two conditions that do not trivially hold. First, the problem of the welfare-maxmising public provider is well-behaved if

$$\begin{aligned} \frac{\partial ^{2}W}{\partial q_{1}^{2}}=-\frac{\left( 16kt-15\beta ^{2}\right) }{16t}<0, \end{aligned}$$

(A7)

which requires $k>15\beta ^{2}/16t$. Second, the Nash equilibrium is locally stable if the Jacobian of the system of first-order conditions is negative definite, which requires

$$\begin{aligned} \frac{\partial ^{2}W}{\partial q_{1}^{2}}\frac{\partial ^{2}\pi _{2}}{ \partial q_{2}^{2}}-\frac{\partial ^{2}\pi _{2}}{\partial q_{2}\partial q_{1} }\frac{\partial ^{2}W}{\partial q_{1}\partial q_{2}}=\frac{k}{16t}\left( 16kt-15\beta ^{2}\right) >0 \end{aligned}$$

(A8)

and

$$\begin{aligned} \left| \begin{array}{ccc} \frac{\partial ^{2}W}{\partial q_{1}^{2}} &{} \frac{\partial ^{2}W}{\partial q_{1}\partial q_{2}} &{} \frac{\partial ^{2}W}{\partial q_{1}\partial q_{3}} \\ \frac{\partial ^{2}\pi _{2}}{\partial q_{2}\partial q_{1}} &{} \frac{\partial ^{2}\pi _{2}}{\partial q_{2}^{2}} &{} \frac{\partial ^{2}\pi _{2}}{\partial q_{2}\partial q_{3}} \\ \frac{\partial ^{2}\pi _{3}}{\partial q_{3}\partial q_{1}} &{} \frac{\partial ^{2}\pi _{3}}{\partial q_{3}\partial q_{2}} &{} \frac{\partial ^{2}\pi _{3}}{ \partial q_{3}^{2}} \end{array} \right| =-\frac{k\left( kt\left( 16kt-23\beta ^{2}\right) +6\beta ^{4}\right) }{16t^{2}}<0. \end{aligned}$$

(A9)

(A8) holds if (A7) holds, while (A9) holds if $kt\left( 16kt-23\beta ^{2}\right) +6\beta ^{4}>0$. Notice that

$$\begin{aligned} \left. kt\left( 16kt-23\beta ^{2}\right) +6\beta ^{4}\right| _{k=\frac{ 15\beta ^{2}}{16t}}=-\frac{3}{2}\beta ^{4}<0 \end{aligned}$$

(A10)

and

$$\begin{aligned} \frac{\partial \left( kt\left( 16kt-23\beta ^{2}\right) +6\beta ^{4}\right) }{\partial k}=t\left( 32kt-23\beta ^{2}\right)>0\text { for }k> \frac{15\beta ^{2}}{16t}, \end{aligned}$$

(A11)

which implies that the condition in (A9) holds if k is above some threshold value higher than $15\beta ^{2}/16t$, which in turn implies that (A7) and (A8) always hold if (A9) holds.

Furthermore, the regulator’s optimal pricing problem (for a given copayment rate) is well-behaved if

$$\begin{aligned} \frac{\partial ^{2}W}{\partial {\overline{p}}^{2}}=-\frac{\left( 2kt-3\beta ^{2}\right) \Theta }{64kt^{2}\left( kt\left( 16kt-23\beta ^{2}\right) +6\beta ^{4}\right) ^{2}}<0, \end{aligned}$$

(A12)

where

$$\begin{aligned} \Theta:&=16kst\left( \beta ^{2}+16kt\right) \left( kt-\beta ^{2}\right) \left( 7\beta ^{2}+4s\left( 2kt-\beta ^{2}\right) \right) \nonumber \\&\quad+49\beta ^{2}\left( 8k^{2}t^{2}\left( 16kt-37\beta ^{2}\right) +\beta ^{4}\left( 205kt-36\beta ^{2}\right) \right) . \end{aligned}$$

(A13)

Assuming that $\Theta >0$, the condition in (A12) holds if $k>3\beta ^{2}/2t$ . Evaluating the numerator in (A9) at $k=3\beta ^{2}/2t$ yields

$$\begin{aligned} \left. kt\left( 16kt-23\beta ^{2}\right) +6\beta ^{4}\right| _{k=\frac{ 3\beta ^{2}}{2t}}=\frac{15}{2}\beta ^{4}>0. \end{aligned}$$

(A14)

Thus, the condition in (A9) always holds if (A12) holds. It remains to show that $\Theta >0$. To do so, we derive

$$\begin{aligned} \frac{\partial ^{3}\Theta }{\partial k^{3}}=768t^{3}\left( 7\beta ^{2}\left( 2s+7\right) +s^{2}\left( 64kt-23\beta ^{2}\right) \right) . \end{aligned}$$

(A15)

Notice that $\partial ^{3}\Theta /\partial k^{3}>0$ if $k>3\beta ^{2}/2t$. This implies that $\partial ^{2}\Theta /\partial k^{2}$ is monotonically increasing in k. Evaluated at the lower bound $k=3\beta ^{2}/2t$, we derive

$$\begin{aligned} \left. \frac{\partial ^{2}\Theta }{\partial k^{2}}\right| _{k=\frac{ 3\beta ^{2}}{2t}}=112t^{2}\beta ^{4}\left( 114s+272s^{2}+245\right) >0. \end{aligned}$$

(A16)

Thus, $\Theta$ is strictly convex for $k>3\beta ^{2}/2t$. Furthermore,

$$\begin{aligned} \left. \frac{\partial \Theta }{\partial k}\right| _{k=\frac{3\beta ^{2}}{ 2t}}=t\beta ^{6}\left( 6944s+10\,336s^{2}+8869\right) >0 \end{aligned}$$

(A17)

and

$$\begin{aligned} \left. \Theta \right| _{k=\frac{3\beta ^{2}}{2t}}=\frac{75}{2 }\beta ^{8}\left( 56s+64s^{2}+49\right) >0. \end{aligned}$$

(A18)

Since $\Theta$ is positive and increasing in k at $k=3\beta ^{2}/2t$, and since $\Theta$ is strictly convex for all $k>3\beta ^{2}/2t$, it follows that $\Theta$ is positive also for all $k>3\beta ^{2}/2t$. Thus, the second-order condition (A12) is satisfied if

$$\begin{aligned} k>{\underline{k}}:=\frac{3\beta ^{2}}{2t}, \end{aligned}$$

(A19)

and this condition ensures that the critical conditions in the quality subgame, (A7)-(A9), are also satisfied.

1.2 B. Proof of proposition 1

(i) Plugging (36) and (37) into (30)–(33), it is easily confirmed that $q_{1}^{**}=q_{2}^{**}=q_{3}^{**}=q^{fb}$ and that $p_{3}^{**}=s^{fb}{\overline{p}}^{fb}$, which implies that $D_{1}=D_{2}=D_{3}$. (ii) In the equilibrium outcome given by (20)–(22), the two private providers have equal quality and price levels, and in turn demand, while demand for the public provider is generally different. Aggregate transportation costs are minimised when each provider has the same demand. Setting $D_{1}=D_{2}=D_{3}$ and solving for $r_{1}$, we find that equal demand across the three providers is induced if ${\overline{p}}$ and s (recall that $r_{1}=$ $s{\overline{p}}$) are set such that

$$\begin{aligned} r_{1}=\frac{\left( 5kt-3\beta ^{2}\right) t+5c\left( 3kt-2\beta ^{2}\right) }{5\left( 3kt-2\beta ^{2}\right) }. \end{aligned}$$

(B1)

This yields the following equilibrium qualities:

$$\begin{aligned} q_{1}^{*}=\beta \frac{45kt-28\beta ^{2}}{45k\left( 3kt-2\beta ^{2}\right) } \end{aligned}$$

(B2)

and

$$\begin{aligned} q_{2}^{*}=q_{3}^{*}=\frac{14\beta }{45k}. \end{aligned}$$

(B3)

Since $D_{1}=D_{2}=D_{3}=1/3$, this means that the average quality is given by

$$\begin{aligned} {\overline{q}}^{*}:=\frac{q_{1}^{*}+q_{2}^{*}+q_{3}^{*}}{3} =\beta \frac{43kt-28\beta ^{2}}{45k\left( 3kt-2\beta ^{2}\right) }. \end{aligned}$$

(B4)

A comparison with the first-best quality yields

$$\begin{aligned} {\overline{q}}^{*}-q^{fb}=-\frac{2\beta \left( kt-\beta ^{2}\right) }{ 45k\left( 3kt-2\beta ^{2}\right) }<0. \end{aligned}$$

(B5)

Thus, when the funding parameters are set such that aggregate mismatch costs are minimised, average quality provision is below the first-best level. Q.E.D.