Effects of price cap regulation on pharmaceutical supply chain under the zero markup drug policy

Abstract

In 2009, China initiated a zero-markup for drugs (ZMD) policy to prevent profit-oriented behaviors among the healthcare providers and alleviate the financial burden on patients. However, overtreatment still prevails because pharmaceutical producers barrage the doctor-patient relationship with promotional activities. This study considered a pharmaceutical supply chain composed of one pharmaceutical producer and one healthcare provider in the context of the ZMD policy, seeking to derive the optimal decisions of the supply chain by considering three scenarios: (1) no promotional activities or price cap regulation; (2) promotional activities but no price cap regulation; and (3) promotional activities and price cap regulation. By comparing the equilibria of these scenarios, we evaluated the impacts of the producers' promotional activities on supply chain members' decisions, profits, and social welfare, as well as the intervention role played by price cap regulation. Our results indicate that the producers' promotional activities contribute to overtreatment, thereby negatively affecting social welfare and patients with low to moderate financial resources. However, the implementation of price cap regulation can effectively reduce the extent of producers' promotional activities, curb overtreatment, and mitigate the associated harm to social welfare.

Appendix

1.1 Proof of Proposition 1

Taking the second order derivative of the producer's profit function ${\pi }_{h}$, obtains $\frac{{d}^{2}{\pi }_{h}}{d{p}_{h}^{2}}=-\frac{2}{1-\delta }<0$. Thus, ${\pi }_{h}$ is strictly concave in ${p}_{h}$. According to the first order condition (FOC), we can get the optimal ${p}_{h}$ from equation (A-1),

$$\frac{d{\pi }_{h}}{d{p}_{h}}=1-\frac{2}{1-\delta }{p}_{h}+\frac{{p}_{l}}{1-\delta }=0.$$

(A-1)

After simplification, we can obtain ${p}_{h}^{b}=\left(1-\delta +{p}_{l}\right)/2$, which, when substituted into the producer's profit function, patients' surplus and social welfare, yields ${\pi }^{b}$, $C{S}^{b}$ and ${W}^{b}$.

1.2 Proof of Property 1

For the patient's surplus in equilibrium $C{S}^{b}$, we can get the first order derivative $\frac{\partial C{S}^{b}}{\partial {p}_{l}}=\frac{2(4-3\delta ){p}_{l}}{8\delta (1-\delta )}-\frac{3}{4}$, $\frac{\partial C{S}^{b}}{\partial \delta }=\frac{3}{8}-\frac{[4-(8-3\delta )\delta ]{p}_{l}^{2}}{8{(1-\delta )}^{2}{\delta }^{2}}$. According to the condition $0<{p}_{l}<\frac{\delta \left(1-\delta \right)}{2-\delta }$, we get $\frac{\partial C{S}^{b}}{\partial {p}_{l}}<-\frac{1}{2\left(2-\delta \right)}<0$ and $\frac{\partial C{S}^{b}}{\partial \delta }>\frac{1}{4-2\delta }>0$.

For the producer's profit in equilibrium ${\pi }^{b}$, we calculate $\frac{\partial {\pi }^{b}}{\partial {p}_{l}}=1-\frac{(4-3\delta ){p}_{l}}{2(1-\delta )\delta }$, $\frac{\partial {\pi }^{b}}{\partial \delta }=-\frac{{\left(1-\delta \right)}^{2}-{p}_{l}^{2}}{4{(1-\delta )}^{2}}$. According to the condition $0<{p}_{l}<\frac{\delta \left(1-\delta \right)}{2-\delta }$, we obtain $\frac{\partial {\pi }^{b}}{\partial {p}_{l}}>\frac{\delta }{4-2\delta }>0$ and $\frac{\partial {\pi }^{b}}{\partial \delta }<0$.

For the social welfare in equilibrium ${W}^{b}$, we get $\frac{\partial {W}^{b}}{\partial {p}_{l}}=\frac{1}{4}-\frac{(4-3\delta ){p}_{l}}{4\delta (1-\delta )}$ and $\frac{\partial {W}^{b}}{\partial \delta }=\frac{1}{8}+\frac{(2-\delta )(2-3\delta ){p}_{l}^{2}}{8{(1-\delta )}^{2}{\delta }^{2}}$. According to the same condition, we find that ${W}^{b}$ first increases and then decreases with ${p}_{l}$, $\frac{1}{4}>\frac{\partial {W}^{b}}{\partial {p}_{l}}>-\frac{1-\delta }{2\left(2-\delta \right)}>0$. When $\delta <\frac{2}{3}$, there is $\frac{\partial {W}^{b}}{\partial \delta }>\frac{1}{8}>0$.

1.3 Proof of Proposition 2

Let Eq. (6) greater than or equal to zero, and compare the results to ${u}_{h}\left(\theta \right)\ge 0$, one can easily justify the proposition.

1.4 Proof of Proposition 3

(1) In the situation of $\eta \le H\left({p}_{h}\right)$, ${\pi }_{h}=({p}_{h}-\eta )(1-\frac{{p}_{h}-{p}_{l}}{1-\delta }+\frac{\eta }{{o}_{d}(1-\delta )})$. We can get the second order derivative $\frac{{\partial }^{2}{\pi }_{h}}{\partial {\eta }^{2}}=-\frac{2}{{o}_{d}\left(1-\delta \right)}<0$, which indicates that the profit function is concave in the promotional effort $\eta$ and the optimal solution exists. According to FOC, we get

$$\eta =\frac{1}{2}\left[\left(1+{o}_{d}\right){p}_{h}-{o}_{d}{p}_{l}-\left(1-\delta \right){o}_{d}\right].$$

(A-2)

Substitute (A-2) into the objective function ${\pi }_{h}$, then take the first derivative with respect to ${p}_{h}$, we obtain,

$$\frac{d{\pi }_{h}}{d{p}_{h}}=\frac{\left(1-{o}_{d}\right)\left[\left(1-{o}_{d}\right){p}_{h}+\left(1-\delta +{p}_{l}\right){o}_{d}\right]}{2{o}_{d}\left(1-\delta \right)}>0.$$

(A-3)

The producer's profit will increase with the branded drug price. Thus, under the condition $\eta \le H\left({p}_{h}\right)$, the optimal solution of ${p}_{h}$ is the boundary solution, satisfying $\eta =H\left({p}_{h}\right)={o}_{d}\left(\delta {p}_{h}-{p}_{l}\right)$. Then we get ${p}_{h}=\frac{\eta +{o}_{d}{p}_{l}}{\delta {o}_{d}}$.

(2) In the situation of $\eta \ge H\left({p}_{h}\right)$, ${\pi }_{h}=({p}_{h}-\eta )(1-{p}_{h})$ and take its first order derivative with respect to $\eta$, yielding

$$\frac{\partial {\pi }_{h}}{\partial \eta }=-\left(1-{p}_{h}\right)<0.$$

(A-4)

The producer's profit will decease with promotional effort. Thus, under the condition $\eta \ge H\left({p}_{h}\right)$, the optimal solution of $\eta$ is the boundary solution, satisfying $\eta =H\left({p}_{h}\right)$. Substitute it into the profit function ${\pi }_{h}$, we can get,

$$\frac{{d}^{2}{\pi }_{h}}{d{p}_{h}^{2}}=-2\left(1-\delta {o}_{d}\right)<0,\frac{d{\pi }_{h}}{d{p}_{h}}=\left(1-\delta {o}_{d}\right)\left(1-2{p}_{h}\right)-{o}_{d}{p}_{l}.$$

(A-5)

When satisfying the FOC $\frac{d{\pi }_{h}}{d{p}_{h}}=0$, we obtain ${p}_{h}=\frac{1-\delta {o}_{d}-{o}_{d}{p}_{l}}{2\left(1-\delta {o}_{d}\right)}$.

Assume that the maximum values of ${\pi }_{h}$ under $\eta \le H\left({p}_{h}\right)$ and $\eta \ge H\left({p}_{h}\right)$ are ${\pi }_{h1}$ and ${\pi }_{h2}$, respectively. Substitute the optimal decisions into profit functions, we have ${\pi }_{h1}\le {\pi }_{h2}$. Therefore, we can get the unique optimal decisions of price and promotional effort.

1.5 Proof of Property 2

Take the first order partial derivative of ${p}_{h}^{e}$ with respect to ${o}_{d}$, $\delta$ and ${p}_{l}$, we obtain $\frac{\partial {p}_{h}^{e}}{\partial {o}_{d}}=-\frac{{p}_{l}}{2{\left(1-\delta {o}_{d}\right)}^{2}}<0$, $\frac{\partial {p}_{h}^{e}}{\partial \delta }=-\frac{{o}_{d}^{2}{p}_{l}}{2{\left(1-\delta {o}_{d}\right)}^{2}}<0$ and $\frac{\partial {p}_{h}^{e}}{\partial {p}_{l}}=-\frac{{o}_{d}}{2\left(1-{o}_{d}\right)}<0$. Then, take the first order partial derivative of ${\eta }^{e}$ with respect to ${o}_{d}$, $\delta$ and ${p}_{l}$, we obtain $\frac{\partial {\eta }^{e}}{\partial {o}_{d}}=\frac{H\left({o}_{d}\right)}{2{\left(1-\delta {o}_{d}\right)}^{2}}$, $\frac{\partial {\eta }^{e}}{\partial \delta }=\frac{{o}_{d}\left({\left(1-{o}_{d}\right)}^{2}-{o}_{d}{p}_{l}\right)}{2{\left(1-\delta {o}_{d}\right)}^{2}}$ and $\frac{\partial {\eta }^{e}}{\partial {p}_{l}}=-\frac{{o}_{d}\left(2-{o}_{d}\right)}{2{\left(1-\delta {o}_{d}\right)}^{2}}<0$. $H\left({o}_{d}\right)=\delta -\delta {o}_{d}\left(\delta -{p}_{l}\right)(2-\delta {o}_{d})-2{p}_{l}$.

According to $\frac{\partial H\left({o}_{d}\right)}{\partial {p}_{l}}=\delta {o}_{d}\left(2-\delta {o}_{d}\right)-2$, we can prove that $H\left({o}_{d}\right)$ is monotonous in range $\left[0, 1\right]$ with respect to ${p}_{l}$, and $H\left({o}_{d}=0,{p}_{l}=0\right)=\delta >0$. When ${o}_{d}=1$, we get,

$$H\left({o}_{d}=1,{p}_{l}\right)=-\left(2+{\delta }^{2}-2\delta \right){p}_{l}+\delta {\left(1-\delta \right)}^{2}.$$

(A-6)

Apparently, $H\left({o}_{d}=1,{p}_{l}\right)$ is negatively related to ${p}_{l}$, $H\left({o}_{d}=1,{p}_{l}=0\right)=\delta {\left(1-\delta \right)}^{2}>0$, $H\left({o}_{d}=1,{p}_{l}=\frac{\delta \left(1-\delta \right)}{2-\delta }\right)=\frac{-{\delta }^{2}\left(1-\delta \right)}{2-\delta }<0$. Thus, there is a $\widehat{{p}_{l}}$ which satisfies $H\left({o}_{d}=1,{p}_{l}=\widehat{{p}_{l}}\right)=0$ reasonable.

1.6 Proof of Property 3

Take the first order partial derivative of $C{S}^{e}$ with respect to ${o}_{d}$, we obtain,

$$\frac{\partial C{S}^{e}}{\partial {o}_{d}}=\frac{\left(1+\delta \right){\left(\delta -{p}_{l}\right)}^{2}}{4{\left(1-\delta {o}_{d}\right)}^{2}}{o}_{d}-\frac{\left(\delta -{p}_{l}\right)\left(1+\delta -2{p}_{l}\right)}{4{\left(1-\delta {o}_{d}\right)}^{2}}.$$

(A-7)

The equation above is always lower than 0 in the range of ${o}_{d}\in \left[0, 1\right)$. We further compare the patient's surplus $C{S}^{e}$ and $C{S}^{b}$, then get,

$$C{S}^{e}\le C{S}^{e}\left({o}_{d}=0\right)=\frac{{p}_{l}^{2}-\delta {p}_{l}}{2\delta }+\frac{1+\delta }{8}<C{S}^{b}.$$

(A-8)

Compare social welfare in different scenarios ${W}^{e}$ and ${W}^{b}$, get the equation,

$$\Delta \left({p}_{l}\right)={W}^{e}-{W}^{b}=\frac{-{\delta }^{2}{o}_{d}^{3}+\left(6\delta -1\right){o}_{d}^{2}-2\left(2+\delta \right){o}_{d}+1}{8\left(1-\delta \right){\left(1-\delta {o}_{d}\right)}^{2}}{p}_{l}^{2}+\frac{\left(1+\delta \right){o}_{d}-{\delta }^{2}{o}_{d}^{2}-1+\delta }{4\left(1-\delta \right)\left(1-\delta {o}_{d}\right)}{p}_{l}-\frac{{\delta }^{2}{o}_{d}}{8\left(1-\delta \right)}.$$

(A-9)

In the range of ${p}_{l}\in [0,\frac{\delta \left(1-\delta \right)}{2-\delta }]$, the conditions $\Delta \left(0\right)<0$ and $\Delta \left(\frac{\delta \left(1-\delta \right)}{2-\delta }\right)<0$ always hold. Because $\Delta \left({p}_{l}\right)$ is a quadratic function of ${p}_{l}$, the extreme point exists, denoted as ${p}_{l}^{*}$,

$${p}_{l}^{*}=-\frac{\left(1-\delta {o}_{d}\right)\left({\delta }^{2}{o}_{d}^{2}-\delta {o}_{d}-\delta -{o}_{d}+1\right)}{{\delta }^{2}{o}_{d}^{3}-6\delta {o}_{d}^{2}+2\delta {o}_{d}+{o}_{d}^{2}+4{o}_{d}-1}.$$

(A-10)

Substitute (A-10) into $\Delta \left({p}_{l}\right)$, we can get the extremum in the domain of definition,

$${\Delta }^{*}\left({p}_{l}^{*}\right)=\frac{{\left[1-\delta +{\delta }^{2}{o}_{d}^{2}-\left(1+\delta \right){o}_{d}\right]}^{2}}{8\left(1-\delta \right)\left[{\delta }^{2}{o}_{d}^{3}-\left(1-6\delta \right){o}_{d}^{2}+2\left(2+\delta \right){o}_{d}-1\right]}-\frac{{\delta }^{2}{o}_{d}}{8\left(1-\delta \right)}.$$

(A-11)

According to (A-11), we define a function: $Den={\delta }^{2}{o}_{d}^{3}-\left(1-6\delta \right){o}_{d}^{2}+2\left(2+\delta \right){o}_{d}-1$. When $Den<0$, ${\Delta }^{*}\left({p}_{l}^{*}\right)<0$ and ${W}^{e}<{W}^{b}$ always hold. When $Den>0$, in the range of $0\le {p}_{l}^{*}\le \frac{\delta \left(1-\delta \right)}{2-\delta }$, ${o}_{d}$ satisfies,

$$\frac{\delta +1-\sqrt{4{\delta }^{3}-3{\delta }^{2}+2\delta +1}}{2{\delta }^{2}}\le {o}_{d}\le \frac{3{\delta }^{2}-4\delta +3-\sqrt{9{\delta }^{4}-32{\delta }^{3}+50{\delta }^{2}-32\delta +9}}{2{\delta }^{2}}.$$

(A-12)

After simplification, we obtain $\delta \le \frac{2}{3}$. Furthermore, when and only when ${\Delta }^{*}\left({p}_{l}^{*}\right)<0$, the relationship ${W}^{e}<{W}^{b}$ holds, that is,

$$\frac{-4{\delta }^{2}+2\delta +1-\sqrt{16{\delta }^{5}-28{\delta }^{4}+24{\delta }^{3}-16{\delta }^{2}+4\delta +1}}{2{\delta }^{2}\left(3-4\delta \right)}\le {o}_{d}\le \frac{-4{\delta }^{2}+2\delta +1+\sqrt{16{\delta }^{5}-28{\delta }^{4}+24{\delta }^{3}-16{\delta }^{2}+4\delta +1}}{2{\delta }^{2}\left(3-4\delta \right)}.$$

(A-13)

When $\delta \le 2/3$, Eq. (A-13) always holds.

Compare the relationship between ${p}_{h}^{b}$ and ${p}_{h}^{e}$, ${p}_{h}^{b}-{p}_{h}^{e}=\frac{{p}_{l}\left(1+{o}_{d}-\delta {o}_{d}\right)-\delta (1-\delta {o}_{d})}{2\left(1-\delta {o}_{d}\right)}$. We find when ${p}_{l}>\frac{\delta (1-\delta {o}_{d})}{1+{o}_{d}-\delta {o}_{d}}$, ${p}_{h}^{b}>{p}_{h}^{e}$ holds.

1.7 Proof of Proposition 4

When $\eta \le H\left({p}_{h}\right)={o}_{d}\left(\delta {p}_{h}-{p}_{l}\right)$, the profit function ${\pi }_{h}$ is monotonically increasing with respect to ${p}_{h}$ based on (A-3). Thus, when $\overline{p }\le \frac{\eta +{o}_{d}{p}_{l}}{\delta {o}_{d}}$, the optimal pricing decision is ${p}_{h}^{c}=\overline{p }$. Meanwhile, the promotional effort satisfies (A-2). When and only when $\left(1+{o}_{d}\right)\overline{p }-{o}_{d}{p}_{l}-\left(1-\delta \right){o}_{d}\ge 0$, the condition $\eta \ge 0$ holds. Therefore, the price cap satisfies $\overline{p}\ge {P }_{e1}=\frac{{o}_{d}\left(1-\delta +{p}_{l}\right)}{1+{o}_{d}}$, that is, when $\overline{p}\le {P }_{e1}$, the optimal promotional effort is ${\eta }^{c}=0$.

When ${p}_{h}^{c}=\overline{p }$ and $\eta =\frac{\left(1+{o}_{d}\right)\overline{p }-{o}_{d}{p}_{l}-\left(1-\delta \right){o}_{d}}{2}>0$, we can obtain the price cap range ${P}_{e1}<\overline{p}\le {P }_{e2}=\frac{{o}_{d}\left(1-\delta -{p}_{l}\right)}{1+{o}_{d}-2\delta {o}_{d}}$ according to $\eta \le H\left({p}_{h}\right)$.

When $\overline{p }<{p}_{h}^{e}$, the optimal pricing decision is ${p}_{h}^{c}=\overline{p }$. At this time, the price cap range satisfies ${P}_{e2}<\overline{p}\le {P }_{e3}=\frac{1-\delta {o}_{d}-{o}_{d}{p}_{l}}{2\left(1-\delta {o}_{d}\right)}$ when the promotional effort is $\eta =H\left({p}_{h}\right)$.

When $\overline{p }>{P}_{e3}$, the optimal decisions of producer under price cap regulation resembles the situation without price cap policy.

1.8 Proof of Property 4

• 1. When $\overline{p}\le {P }_{e1}$, we get $\frac{dC{S}^{c}}{d\overline{p} }=\frac{\overline{p }-{p}_{l}}{1-\delta }-1$. Thus, $C{S}^{c}$ monotonically decreases with the price cap $\overline{p }$ in the range of $\overline{p }<1-\delta +{p}_{l}$, and monotonically increases in the range of $\overline{p }>1-\delta +{p}_{l}$. Due to ${P}_{e1}=\frac{{o}_{d}\left(1-\delta +{p}_{l}\right)}{1+{o}_{d}}<1-\delta +{p}_{l}$, when $\overline{p}\le {P }_{e1}$, there is $\frac{dC{S}^{c}}{d\overline{p} }<0$. Similarly, according to $\frac{d{W}^{c}}{d\overline{p} }=-\frac{\overline{p }-{p}_{l}}{1-\delta }$, $C{S}^{c}$ is monotonically increasing when $\overline{p }<{p}_{l}$ and monotonically decreasing when $\overline{p }>{p}_{l}$. Due to $\overline{p}\ge {p }_{h}>{p}_{l}$, when $\overline{p}\le {P }_{e1}$, there is $\frac{d{W}^{c}}{d\overline{p} }<0$.

• 2. When ${P}_{e1}<\overline{p}\le {P }_{e2}$, we get $\frac{dC{S}^{c}}{d\overline{p} }=-\frac{\left(1+3{o}_{d}\right)\left(1-{o}_{d}\right)}{4\left(1-\delta \right){o}_{d}^{2}}\overline{p }+\frac{\left(1-3{o}_{d}\right)\left(1-\delta +{p}_{l}\right)}{4\left(1-\delta \right){o}_{d}}$. Therefore, $C{S}^{c}$ is monotonically increasing in the range of $\overline{p }<\frac{\left(1-3{o}_{d}\right)\left(1-\delta +{p}_{l}\right)}{\left(1+3{o}_{d}\right)\left(1-{o}_{d}\right)}$ and monotonically decreasing in the range of $\overline{p }>\frac{\left(1-3{o}_{d}\right)\left(1-\delta +{p}_{l}\right)}{\left(1+3{o}_{d}\right)\left(1-{o}_{d}\right)}$. Because ${P}_{e1}>\frac{\left(1-3{o}_{d}\right)\left(1-\delta +{p}_{l}\right)}{\left(1+3{o}_{d}\right)\left(1-{o}_{d}\right)}$, when ${P}_{e1}<\overline{p}\le {P }_{e2}$, $\frac{dC{S}^{c}}{d\overline{p} }<0$. Similarly, according to $\frac{d{W}^{c}}{d\overline{p} }=-\frac{{o}_{d}^{3}+3{o}_{d}^{2}-{o}_{d}+1}{4\left(1-\delta \right){o}_{d}^{2}}\overline{p }+\frac{\left(1-\delta +{p}_{l}\right){o}_{d}^{2}+2{p}_{l}{o}_{d}+1-\delta -{p}_{l}}{4\left(1-\delta \right){o}_{d}}$, ${W}^{c}$ is monotonically increasing when $\overline{p }<\frac{{o}_{d}\left(\left(1-\delta +{p}_{l}\right){o}_{d}^{2}+2{p}_{l}{o}_{d}+1-\delta -{p}_{l}\right)}{{o}_{d}^{3}+2{o}_{d}^{2}-{o}_{d}+1}$ and monotonically decreasing when $\overline{p }>\frac{{o}_{d}\left(\left(1-\delta +{p}_{l}\right){o}_{d}^{2}+2{p}_{l}{o}_{d}+1-\delta -{p}_{l}\right)}{{o}_{d}^{3}+2{o}_{d}^{2}-{o}_{d}+1}$. Because of ${P}_{e1}<{P}_{e2}<{P}_{e3}$, the inequality ${P}_{e1}>\frac{{o}_{d}\left(\left(1-\delta +{p}_{l}\right){o}_{d}^{2}+2{p}_{l}{o}_{d}+1-\delta -{p}_{l}\right)}{{o}_{d}^{3}+2{o}_{d}^{2}-{o}_{d}+1}$ always holds. Thus, when ${P}_{e1}<\overline{p}\le {P }_{e2}$, $\frac{d{W}^{c}}{d\overline{p} }<0$.

• 3. When ${P}_{e2}<\overline{p}\le {P }_{e3}$, $\frac{dC{S}^{c}}{d\overline{p} }=\left(1+\delta \right)\overline{p }-1-{p}_{l}$. Then we can obtain that $C{S}^{c}$ is monotonically decreasing in the range of $\overline{p }<\frac{1+{p}_{l}}{1+\delta }$ and monotonically increasing in the range of $\overline{p }>\frac{1+{p}_{l}}{1+\delta }$. Due to ${P}_{e3}<\frac{1+{p}_{l}}{1+\delta }$, when ${P}_{e2}<\overline{p}\le {P }_{e3}$ $\frac{dC{S}^{c}}{d\overline{p} }<0$. Similarly, based on $\frac{d{W}^{c}}{d\overline{p} }=-\frac{{{\delta }^{2}{o}_{d}+\left(1-\delta \right)}^{2}}{1-\delta }\overline{p }+\frac{\delta {o}_{d}{p}_{l}}{\left(1-\delta \right)}$, we get that ${W}^{c}$ is monotonically increasing in the range of $\overline{p }<\frac{\delta {o}_{d}{p}_{l}}{{\delta }^{2}{o}_{d}+{\left(1-\delta \right)}^{2}}$ and monotonically decreasing in the range of $\overline{p }>\frac{\delta {o}_{d}{p}_{l}}{{\delta }^{2}{o}_{d}+{\left(1-\delta \right)}^{2}}$. Due to ${P}_{e1}<{P}_{e2}<{P}_{e3}$, ${P}_{e2}>\frac{\delta {o}_{d}{p}_{l}}{{\delta }^{2}{o}_{d}+{\left(1-\delta \right)}^{2}}$ always holds. Thus, when ${P}_{e2}<\overline{p}\le {P }_{e3}$, $\frac{d{W}^{c}}{d\overline{p} }<0$.