Mixed provision of health care services with double coverage

Abstract

This paper studies the effect of two alternative regulations of a health care market characterized by mixed provision. Health care is considered as a bundle of mild and severe illness treatments. A free national health service (NHS) offering universal coverage and preferred for severe illnesses treatments, but congested for mild ones, competes with a private health service which is not congested and preferred for mild illnesses care. Either an exclusive subscription regulation, where users are forced to demand health care from only one service; or a non-exclusive subscription regulation that allows choosing for each kind of illness a different health service can be implemented. The move from an exclusive to a non-exclusive regulation implies some users shifting to the private provider for mild illnesses only. This leads to an overall reduction of congestion in the NHS that always benefits users and leads to a less costly NHS under some conditions. Under the latter regulation the private service specializes in mild illnesses with higher profits, while all severe cases are treated by the NHS. Finally, social welfare is also higher under the non-exclusive regulation, even assuming a positive social cost of raising funds to cover the public health care, if such inefficiency is moderate or the NHS is less costly.

Appendix

Equilibrium expressions

In this section of the Appendix we will present the equilibrium expressions corresponding to both the ES and the NES regulation for the private and the regulated equilibrium.

ES regulation

Private equilibrium

Equilibrium price,

$p_{B}^{*}=\frac{1+k+d-\rho (w-c)}{2}.$

Demands at equilibrium,

$\alpha _A ^{*}=\frac{3+k+d+\rho (c+w)}{2(2+k)}, (1-\alpha _A ^{*})=\frac{1+k-d-\rho (c+w)}{2(2+k)}.$

Equilibrium profits,

$\pi _{A}^{*}=I^{*}-\frac{(d+\rho c)(3+k+d+\rho (c+w))}{2(2+k)}$, and $I^{*}= \frac{(d+\rho c)(3+k+d+\rho (c+w))}{2(2+k)};$ $\pi _{B}^{*}=\frac{(1+k-d-\rho (c+w))^2}{4(2+k)}$

Consumer surplus,

$$\begin{aligned}&cs^{*}=v-\frac{1}{2}+ \rho w_B-p_{B}^{*}(1-\alpha _A ^{*})+(1+\rho w)\alpha _A ^{*}-(1+k) (\alpha _A ^{*})^2\\&=v-\frac{1}{2}{+} \rho w_B\\&\quad {+} \frac{{(1{-}d)^2-k (2{+}k) (5{+}2 k){+}2 d (\rho (c{+}w)\!-\!k (3{+}k)){+}2 \rho ((7{+}k (5{+}k)) w{-}c k (3{+}k)){-}2 c \rho {+}\rho ^2 (c{+}w)^2}}{4 (2{+}k)^2} \end{aligned}$$

Social welfare,

$$\begin{aligned} sw^{*}= & {} v {-}\frac{1}{2}{-}d{+} \rho \left( w_{B}{-}c\right) {+} \left( 1{+}\rho w\right) \alpha _{A}^{*}{-}\left( 1{+}k\right) \left( \alpha _A^{*}\right) ^{2} {-}\left( \delta {-}1\right) \left( d{+}\rho c\right) \alpha _A^{*}\\= & {} v {-}\frac{1}{2}{-}d{+} \rho \left( w_{B}{-}c\right) {+}\frac{\left( 1{+}\rho w{-}\left( \delta {-}1\right) \left( d{+}\rho c\right) \right) \left( 3{+}k{+}d{+} \rho \left( c{+}w\right) \right) }{2 \left( 2{+}k\right) }\nonumber \\&-\,\frac{\left( 1+k\right) \left( 3+k+d+\rho \left( c+w\right) \right) ^2}{4 \left( 2+k\right) ^2} \end{aligned}$$

Regulated equilibrium.

Equilibrium price,

$p_{B}^{r}=d+ \rho c$.

Demands at equilibrium, $\alpha _A ^{r}=\frac{1+d+\rho (c+w)}{2+k}$, $(1-\alpha _A ^{r })=\frac{1+k-d-\rho (c+w)}{(2+k)}$.

Equilibrium profits, $\pi _{A}^{r }=I^r-\frac{(d+c \rho ) (1+d+\rho (c+w))}{2+k}$, and $I^{r}= \frac{(d+\rho c)(1+d+\rho (c+w))}{2+k}$; $\pi _{B}^{r }=0$

Consumer surplus,

$$\begin{aligned} cs^{r}= & {} v{-}\frac{1}{2}{+} \rho w_B{-}p_{B}^{r }(1{-}\alpha _A ^{r }){+}(1{+}\rho w)\alpha _A ^{r }{-}(1{+}k) (\alpha _A ^{r })^2\nonumber \\= & {} v{-}\frac{1}{2}{+} \rho w_B {+}\frac{(1{-}d)^2{+}d (2 \rho (c{+}w){-}k (k{+}4)){+}\rho \left( 2 w {-}c (k (k{+}4){+}2){+}\rho (c{+}w)^2\right) }{(2{+}k)^2} \end{aligned}$$

Social welfare,

$$\begin{aligned} sw^{r }= & {} v {-}\frac{1}{2}{-}d{+} \rho \left( w_{B}{-}c\right) {+} \left( 1{+}\rho w\right) \alpha _{A}^{r }{-}\left( 1{+}k\right) \left( \alpha _A^{r }\right) ^{2} {-}\left( \delta {-}1\right) \left( d{+} \rho c\right) \alpha _A^{r}\\= & {} v {-}\frac{1}{2}{-}d{+} \rho \left( w_{B}{-}c\right) {+}\frac{\left( 1{+}\rho w{-}\left( \delta {-}1\right) \left( d{+}\rho c\right) \right) \left( 1{+}d{+} \rho \left( c{+}w\right) \right) }{\left( 2{+}k\right) }\nonumber \\&-\,\frac{\left( 1+k\right) \left( 1+d+\rho \left( c+w\right) \right) ^2}{\left( 2+k\right) ^2} \end{aligned}$$

NES regulation

Private equilibrium

Equilibrium price,

${\bar{p}}_{B}^{*}=\frac{1+k+d}{2}$.

Demands at equilibrium,

$\bar{\alpha _A}^{*}=\frac{3+k+d}{2(2+k)}$, $(1-\bar{\alpha _A}^{*})=\frac{1+k-d}{2(2+k)}$.

Equilibrium profits,

${\bar{\pi }}_{A}^{*} =\bar{I}^{*}-\frac{d(3+k+d)}{2(2+k)}- \rho c$, and $\bar{I}^{*}=\frac{d(3+k+d)}{2(2+k)}+\rho c;$ ${\bar{\pi }}_{B}^{*} =\frac{(1+k-d)^2}{4(2+k)}$

Consumer surplus,

$$\begin{aligned} \bar{cs}^{*}= & {} v-\frac{1}{2}+ \rho w_A-{\bar{p}}_{B}^{*}(1-{\bar{\alpha }}_A ^{*})+{\bar{\alpha }}_A ^{*}-(1+k) ({\bar{\alpha }}_A ^{*})^2\\= & {} v{-}\frac{1}{2}{+} \rho w_A {-}\frac{(1{+}k{-}d) (1{+}k{+}d)}{4 (2{+}k)}{+}\frac{3{+}k{+}d}{2(2{+} k)}{-}\frac{(1{+}k) (3{+}k{+}d)^2}{4 (2{+}k)^2} \end{aligned}$$

Social welfare,

$$\begin{aligned} \bar{sw}^{*}= & {} v -\frac{1}{2}-d+ \rho (w_{A}-c) + {\bar{\alpha }}_{A}^{*}-(1+k) ({\bar{\alpha }}_A^{*})^{2}-(\delta -1)(d{\bar{\alpha }}_A^{*}+\rho c)\\= & {} v -\frac{1}{2}-d+ \rho (w_{A}-\delta c)+\frac{(1-(\delta -1)d)(3+k+d)}{2(2+ k)}\\&-\,\frac{(1+k) (3+k+d)^2}{4 (2+k)^2} \end{aligned}$$

Regulated equilibrium.

Equilibrium price,

${\bar{p}}_{B}^{r}=d$.

Demands at equilibrium,

${\bar{\alpha }}_A ^{r}=\frac{1+d}{2+k}$, $(1-{\bar{\alpha }}_A ^{r })=\frac{1+k-d}{(2+k)}$.

Equilibrium profits,

${\bar{\pi }}_{A}^{r }=I-\frac{d (1+d)}{2+k}- \rho c$, and $\bar{I}^{r}=\frac{d(1+d)}{(2+k)}+\rho c$; ${\bar{\pi }}_{B}^{r }=0$

Consumer surplus,

$$\begin{aligned} \bar{cs}^{r }= & {} v-\frac{1}{2}+ \rho w_A-{\bar{p}}_{B}^{r }(1-{\bar{\alpha }}_A ^{r })+{\bar{\alpha }}_A ^{r }-(1+k) ({\bar{\alpha }}_A ^{r })^2\\= & {} v-\frac{1}{2}+ \rho w_A -d+\frac{(1+d)^2}{(2+k)^2} \end{aligned}$$

Social welfare,

$$\begin{aligned} \bar{sw}^{r }= & {} v -\frac{1}{2}-d+ \rho (w_{A}-c) + {\bar{\alpha }}_{A}^{r }-(1+k) ({\bar{\alpha }}_A^{r })^{2}-(\delta -1)(d{\bar{\alpha }}_A^{r}+\rho c)\\= & {} v {-}\frac{1}{2}{+} \rho (w_{A}{-}c){+}\frac{(1{+}d)(1{-}d(1{+}k)){-}(2{+}k)(d(1{+}d){+}(2{+}k)c \rho )(\delta -1)}{(2+k)^2} \end{aligned}$$

Proof of Proposition 1

Throughout the proof, the most restrictive conditions to ensure positive demands at equilibrium are used. Those conditions read $0<w<\frac{ 1+k-d}{\rho }-c$ and also $d<1+k$.

• We first prove that consumer surplus is greater under the NES regulation:

Notice that $\overline{cs}^{*}>cs^{*}$ iff

$2 \left( c (1+k (k+3)-d)-w (7+k (5+k)+d)+2 (2+k)^2 w\right) -\rho (c+w)^2>0$

The latter inequality holds if $w<\frac{ 2(1+ k (k+3)-d)}{\rho }-c.$ Since $\frac{ 1+k-d}{\rho }-c<\frac{ 2(1+ k (k+3)-d)}{\rho }-c$ then $\overline{cs}^{*}>cs^{*}$. That is, since the condition for positive demands is stronger then the result is proven.

• National Health Service profits relative performance:

We have that ${\bar{\pi }}_{A}^{*}>\pi _{A}^{*}$ iff $\frac{d(c+w)-c(1+k-d-\rho (c+w))}{2 (2+k)}>0$That inequality holds if $d > \hat{d}=\frac{c( 1+k -\rho (c+w))}{ w+2c}$, where $\hat{d}$ is positive. Therefore, the conclusion is that the NHS is less costly under the NES regulation if the marginal cost of mild illness treatments is large enough.

• The private Health service is better off under the NES regulation.

By inspection, ${\bar{\pi }}_{B}^{*} =\frac{(1+k-d)^2}{4(2+k)}>\pi _{B}^{*}=\frac{(1+k-d-\rho (c+w))^2}{4(2+k)}$

• Social welfare is larger under the NES regulation.

• (i) For the case of no inefficiency, i.e. for $\delta =1$, $\overline{sw}^{*}>sw^{*}$ iff

  $c^2 (1+k) \rho +2 c \left( 1+(d+3) k+k^2+d\right) +2 w (3+2 k (k+3)-d-c \rho )-(3+k) \rho w^2>0.$

  The inequality is satisfied for $w^-<w<w^+$, where $w^-$ and $w^+$ are the lower and upper roots for the expression. It happens that $w^-<0$ and $\frac{ 1+k-d}{\rho }-c<w^+$, then $\overline{sw}^{*}>sw^{*}$ for the conditions that ensure positive demands.

• (ii) For the inefficiency case, i.e. for $\delta >1$, $\overline{sw}^{*}>sw^{*}$ when

$(c+w)(6+(3+k)(4k-2d-\rho (c+w)))+2(2+k)(2c+w)(d-\hat{d})\delta >0$

But note that the NHS is less costly under the NES if $d>\hat{d}$ and that $6+(3+k)(4k-2d-\rho (c+w))$ is always positive using the upper limits of w and d indicated in the text. Therefore, if $d>\hat{d}$ then $\overline{sw}^{*}>sw^{*}$ for all $\delta \ge 1$.

Finally, when if $d<\hat{d}$, the second term is negative and $\overline{sw}^{*}>sw^{*}$ when

$\delta <\hat{\delta } \equiv \frac{(c+w)(6+(3+k)(4k-2d-\rho (c+w)))}{2(2+k)(2c+w)(\hat{d}-d)}$.

Regarding the difference of welfare for the regulated case we have a similar proof as above where now the expression required to be positive is

$(c+w)((1+k)^2 +1-2d-\rho (c+w)))+(2+k)(2c+w)(d-\hat{d})\delta >0$.

Therefore, welfare is larger under the regulated NES if either $\delta =1$, or for $d>\hat{d}$ regardless the size of $\delta$, or for $d< \hat{d}$ when $\delta < \hat{\delta ^{r}}$ where $\hat{\delta ^{r}} \equiv \frac{(c+w)((1+k)^2 +1-2d-\rho (c+w)))}{(2+k)(2c+w)(\hat{d}-d)}$ and $\hat{\delta ^{r}}<\hat{\delta }$.