Optimal Cooperation of Medical Care and Nursing Care in a Two-Region Spatial Model

Abstract

This paper presents an analysis of the optimal cooperation of medical care service and nursing care service under two regions with an asymmetric density of patients. We apply Aiura and Sanjo (2010) by introducing cooperation between hospitals and nursing care facilities and analyze the effects of cooperation on the equilibrium medical service level, nursing care service level, optimal medical and nursing care fee, and social welfare. Results of the analysis show that the introduction of cooperation between hospitals and nursing care facilities increases social welfare, although it decreases the medical care service and nursing care service level in equilibrium.

Appendix

1.1 Proof of $\bar{x}^{**}\ngtr 0$

Substituting (15.17) for (15.3), $\bar{x}^{**}$ in equilibrium is given as shown below.

$$\begin{aligned} \bar{x}^{**}= & {} \frac{1}{2}\left( 1-\frac{1}{p}\right) +\left( \frac{\alpha \left( 3\alpha \left( 1-p\right) +4pt\beta \right) }{p\left( 3\alpha -2t\beta \right) \left( \alpha -2t\beta \right) }\right) >0 \end{aligned}$$

(15.58)

Now we assume that $\alpha$ is sufficiently large. When $\alpha$ is sufficiently large, p is larger than 1. However, (15.58) is not satisfied because the range of p is from 0 to 1. Consequently, ${\bar{x}}^{**}$ is not positive.

1.2 Proof of ${\tilde{x}}^{*}\le 0$

Substituting (15.48), (15.49), (15.50), and (15.51) for (15.52), we derive the $\tilde{x}^{*}$ in equilibrium as follows.

$$\begin{aligned} \tilde{x}^{*}=\frac{1}{2}\left( 1-\frac{1}{p}\right) -\frac{\left( p\theta \gamma ^{2}-2t\alpha \delta \right) }{\varGamma }\left[ \theta \left( 1-p\right) 2t\delta \left( 2t\beta +p\theta \alpha \right) \right] \end{aligned}$$

(15.59)

Because $\tilde{x}^{*}$ is not positive, the following inequality must hold.

$$\begin{aligned} \frac{1}{2}\left( 1-\frac{1}{p}\right) -\frac{\left( p\theta \gamma ^{2}-2t\alpha \delta \right) }{\varGamma }\left[ \theta \left( 1-p\right) 2t\delta \left( 2t\beta +p\theta \alpha \right) \right] <0 \end{aligned}$$

(15.60)

Simplifying (15.60), we obtain the simplified inequality as follows.

$$\begin{aligned} \frac{1}{2}\left( 1-\frac{1}{p}\right) -\frac{\left( p\theta \gamma ^{2}-2t\alpha \delta \right) }{\varGamma }\left[ \theta \left( 1-p\right) 2t\delta \left( 2t\beta +p\theta \alpha \right) \right]<0\Leftrightarrow -p\theta \alpha <2\beta t \end{aligned}$$

(15.61)

Now $-p\theta \alpha <2\beta t$ always holds because we assume that p, $\theta$, $\alpha$, $\beta$, and t are positive. Consequently, we can prove that $\tilde{x}^{*}$ is not positive under $\tilde{q}_{1}^{*}$, $\tilde{q}_{2}^{*}$, $\tilde{w}_{1}^{*}$, and $\tilde{w}_{2}^{*}$.