Generic substitution policy, an incentive approach

Abstract

Generic substitution policy has been adopted in several countries in order to control health expenditures. Using a model based on incentives, this work aims to analyze the response of doctors and pharmaceutical companies to the implementation of this policy. It is shown that after the implementation of GSP, the effort of doctor’s convincing the patient to take generics increase or decrease depending on his level of concern for patient well-being; pharmaceutical companies decrease the amount of detailing and the market share of generics tends to increase.

Appendices

Appendix: Proofs

1.1 Observation

We must show that there exist two functions f and g such that one can express the joint distribution of prescription and sales signals as $P(i,j)=f(i,e)g(i,j)$, for all $i=B,G$ and $j=B,G$. Indeed, let

$$\begin{aligned} f\left( i,j\right) =\left\{ \begin{array}{cc} e &\quad {\text {if }}\,\,i=G \\ 1-e & \quad {\text {if }}\,\,i=B \end{array} \right. \end{aligned}$$

and let

$$\begin{aligned} g\left( i,j\right) =\left\{ \begin{array}{cc} 1 &\; \; {\text {if }}\,\,i=G\,\,{\text { and }}\,\,j=G \\ 0 &\; {\text {if} }\,\,i=G\,\,{\text { and }}\,\,j=B \\ \varphi &\; \; {\text {if }}\,\,i=B\,\,{\text { and }}\,\,j=G \\ 1-\varphi &\; \;{\text {if }}\,\,i=B\,\,{\text{ and }}\,\,j=B \end{array} \right. \end{aligned}$$

Then

$$\begin{aligned} P(i,j)=f(i,e)g(i,j)=\left\{ \begin{array}{ll} e\cdot 1=e &\quad {\text {if }}\,\,i=G\,\,{\text { and }}\,\,j=G \\ e\cdot 0=0 &\quad {\text {if }}\,\,i=G\,\,{\text { and }}\,\,j=B \\ \left( 1-e\right) \varphi &\quad {\text {if }}\,\,i=B\,\,{\text { and }}\,\,j=G \\ \left( 1-e\right) \left( 1-\varphi \right) &\quad {\text {if }}\,\,i=B\,\,{\text { and }}\,\,j=B \end{array} \right. , \end{aligned}$$

as given in the text.

The substitution policy is implemented

Lemma 1

The pharma’s objective function $EP^{S}(D,e)=(1-e)\left[ \left( 1-\varphi \right) \pi -D)\right]$.

If $e=0,W-D+\left( 1-\varphi \right) \Delta \le 0$ , then $EP^{S}(D,0)=\left( 1-\varphi \right) \pi -D.$

If $e=1$, $W-D+\left( 1-\varphi \right) \Delta \ge 1$ , then $EP^{S}(D,1)=0$ .

After substituting the doctor’s effort $e^{S*}=W-D+\left( 1-\varphi \right) \Delta ,$

we can write

$$\begin{aligned}&EP^{S}(D,e^{S}\left( D\right) ) \\&=\left\{ \begin{array}{ll} 0 &\quad {\text {if }}\,\,D\le W+\left( 1-\varphi \right) \Delta -1 \\ (1-\left( W-D+\left( 1-\varphi \right) \Delta \right) )\left[ \left( 1-\varphi \right) \pi -D)\right] & \quad {\text {if }}\,\,W+\left( 1-\varphi \right) \Delta -1\\ &\quad <D<W+\left( 1-\varphi \right) \Delta \\ \left( 1-\varphi \right) \pi -D & \quad{\text {if }}\,\,D\ge W+\left( 1-\varphi \right) \Delta \end{array} \right. \end{aligned}$$

From FOC we get $D^{S*}=\frac{1}{2}\left[ \left( 1-\varphi \right) \left( \pi +\Delta \right) +W-1\right]$ and $D^{S*}>0.$

For this solution to be interior we need

• (i) $W+\left( 1-\varphi \right) \Delta -1<D^{S*}<W+\left( 1-\varphi \right) \Delta$

• (ii) $D^{S*}=\frac{1}{2}\left[ \left( 1-\varphi \right) \left( \pi +\Delta \right) +W-1\right] >0.$

Analyzing these inequalities

$$\begin{aligned} \begin{array}{l} (i.a)\, D^{S*}<W+\left( 1-\varphi \right) \Delta \\ \frac{1}{2}\left[ \left( 1-\varphi \right) \left( \pi +\Delta \right) +W-1 \right]<W+\left( 1-\varphi \right) \Delta \\ \frac{1}{2}\left( 1-\varphi \right) \left( \pi +\Delta \right) +\frac{1}{2} W-\frac{1}{2}<W+\left( 1-\varphi \right) \Delta \\ \frac{1}{2}\left( 1-\varphi \right) \pi +\frac{1}{2}\left( 1-\varphi \right) \Delta +\frac{1}{2}W-\frac{1}{2}<W+\left( 1-\varphi \right) \Delta \\ \frac{1}{2}\left( 1-\varphi \right) \pi -\frac{1}{2}<\frac{1}{2}W+\frac{1}{2 }\left( 1-\varphi \right) \Delta \\ \left( 1-\varphi \right) \pi -1<W+\left( 1-\varphi \right) \Delta \\ \left( 1-\varphi \right) \left( \pi -\Delta \right)<W+1\\ \left( 1-\varphi \right) \left( \pi -\Delta \right) -1<W\\ W>\left( 1-\varphi \right) \left( \pi -\Delta \right) -1\\ (i.b)\,W+\left( 1-\varphi \right) \Delta -1<D^{S*}\\ W+\left( 1-\varphi \right) \Delta -1<\frac{1}{2}\left[ \left( 1-\varphi \right) \left( \pi +\Delta \right) +W-1\right] \\ W+\left( 1-\varphi \right) \Delta -1<\frac{1}{2}\left( 1-\varphi \right) \left( \pi +\Delta \right) +\frac{1}{2}W-\frac{1}{2}\\ \frac{1}{2}W+\left( 1-\varphi \right) \Delta -\frac{1}{2}<\frac{1}{2}\left( 1-\varphi \right) \pi +\frac{1}{2}\left( 1-\varphi \right) \Delta \\ \frac{1}{2}W+\frac{1}{2}\left( 1-\varphi \right) \Delta -\frac{1}{2}<\frac{1 }{2}\left( 1-\varphi \right) \pi \\ W+\left( 1-\varphi \right) \Delta -1<\left( 1-\varphi \right) \pi \\ W<\left( 1-\varphi \right) \left( \pi -\Delta \right) +1\\ \end{array} \end{aligned}$$

$$\begin{aligned} (iii) \,D^{S*}=\frac{1}{2}[\left( 1-\varphi \right) \left( \pi +\Delta \right) +W-1]>0\Leftrightarrow W>1-\left( 1-\varphi \right) \left( \pi +\Delta \right) \end{aligned}$$

Therefore, we can deduce the following:

$D^{S*}$ is interior (which implies that $e^{S*}$ is also interior) if and only if

$$\begin{aligned} \max \{0;\left( 1-\varphi \right) \left( \pi -\Delta \right) -1;1-\left( 1-\varphi \right) \left( \pi +\Delta \right) \}<W<\left( 1-\varphi \right) \left( \pi -\Delta \right) +1. \end{aligned}$$

Moreover, from the no-GSP conditions to ensure interior solutions, we know that

$$\begin{aligned} max\{0;1-(\pi +\Delta );(\pi -\Delta )-1\}<W<\pi -\Delta +1 \end{aligned}.$$

Thus, the conditions that guarantee interior solutions in no-GSP and GSP can be stated as it follows:

$$\begin{aligned} max\{0;\left( 1-\varphi \right) \left( \pi -\Delta \right) -1;(\pi -\Delta )-1;1-\left( 1-\varphi \right) \left( \pi +\Delta \right) ;1-(\pi +\Delta )\}\\<W<min\{(\pi -\Delta )+1; \end{aligned}$$

$\left( 1-\varphi \right) \left( \pi -\Delta \right) +1\}.$

Let’s take

$$\begin{aligned} \begin{array}{l} A=\left( 1-\varphi \right) \left( \pi -\Delta \right) -1\\ B=(\pi -\Delta )-1\\ C=1-\left( 1-\varphi \right) \left( \pi +\Delta \right) \\ D=1-(\pi +\Delta )\\ E=(\pi -\Delta )+1\\ F=\left( 1-\varphi \right) \left( \pi -\Delta \right) +1\\ \end{array} \end{aligned}$$

So we have $max\{0;A;B;C;D\}<W<min\{E;F\}$

We next analyze these conditions.

• (A) $min\{E;F\}=F=\left( 1-\varphi \right) \left( \pi -\Delta \right) +1$

• (B) $max\{0;A;B;C;D\}=max\{0;A;B;C\}$ because $C>D$.

Case 1 : Suppose $\Delta \ge \pi ,$ then $max\{0;A;B;C\}=max\{0;C\}$ because $A<0$ and $B<0$.

$C>0$ if $1-\left( 1-\varphi \right) \left( \pi +\Delta \right) >0\Leftrightarrow (1-\varphi )\pi +(1-\varphi )\Delta<1\Leftrightarrow \pi < \dfrac{1}{1-\varphi }-\Delta .$

When $C>0$ then $max\{0;C\}=C.$

Case 2: Suppose $\Delta <\pi ,$ then $max\{0;A;B;C\}=max\{0;B;C\}$ because $B>A$.

Note that $B=(\pi -\Delta )-1>0\Leftrightarrow \pi >\Delta +1.$

$$\begin{aligned} \qquad C=1-\left( 1-\varphi \right) \left( \pi +\Delta \right)>0\Leftrightarrow \pi >\dfrac{1}{1-\varphi }-\Delta . \end{aligned}$$

Case 2.1 When $B<0$ and $C<0$.

Then, $max\{0;B;C\}=0.$

Case 2.2 When $B<0$ and $C>0$.

Then, $max\{0;B;C\}=C.$

Case 2.3 When $B>0$ and $C>0$.

Then, $max\{0;B;C\}=C$ if $\pi <\dfrac{2+\varphi \Delta }{2-\varphi }$

or $max\{0;B;C\}=B$ if $\pi >\dfrac{2+\varphi \Delta }{2-\varphi }.$

Summing up, we get Lemma 1

• (1) Suppose $\Delta \ge \pi .$

• (1.1) If $\pi <\dfrac{1}{1-\varphi }-\Delta ,$ then $C<W<F.$

• (1.2) If $\pi >\dfrac{1}{1-\varphi }-\Delta ,$ then $0<W<F.$

• (2) Suppose $\Delta <\pi .$

• (2.1) If $\pi <\dfrac{1}{1-\varphi }-\Delta$ then $0<W<F.$

• (2.2) If $\dfrac{1}{1-\varphi }-\Delta<\pi <\Delta +1$ then $C<W<F.$

• (2.3) If $\pi >\Delta +1$ and $\pi >\dfrac{2+\varphi \Delta }{ 2-\varphi }$, then $B<W<F.$

• (2.4) If $\pi >\Delta +1$ and $\pi <\dfrac{2+\varphi \Delta }{ 2-\varphi }$, then $C<W<F$

Lemma 2

We have $\Pr ^{N}\left( G\right) =e^{*N}$ and

$$\begin{aligned} \Pr ^{S}\left( G\right) =P(GG)+P(BG)=e^{*S}+\left( 1-e^{*S}\right) \varphi =(1-\varphi )e^{*S}+\varphi . \end{aligned}$$

Moreover, $\ e^{N}(D)=W+\Delta -D,$ $e^{S*}=W-D+(1-\varphi )\Delta$ and D is fixed $\overline{D}=D^{N*}=\frac{1}{2}(\pi +W+\Delta -1)$

$$\begin{aligned} \begin{array}{l} \Pr ^{S}\left( G\right)<\Pr ^{N}\left( G\right) \Leftrightarrow (1-\varphi )e^{*S}+\varphi<e^{*N}\\ \Leftrightarrow (1-\varphi )(W-\overline{D}+(1-\varphi )\Delta )+\varphi<W+\Delta -\overline{D}\Leftrightarrow \\ \Leftrightarrow (W-\overline{D})(1-\varphi )-(W-\overline{D})+(1-\varphi )^{2}\Delta -\Delta +\varphi<0\Leftrightarrow \\ \Leftrightarrow (W-\overline{D})(-\varphi )+(\varphi (\varphi -2))\Delta +\varphi<0\\ \Leftrightarrow (W-\overline{D})+(2-\varphi )\Delta <1\\ \Leftrightarrow W>1-(2-\varphi )\Delta +\overline{D}\\ \end{array} \end{aligned}$$

Proposition 5

Note that $e^{N*}=\frac{1}{2}+\frac{1}{2}(W+\Delta -\pi )$ and $e^{S*}=\frac{1}{2}\left[ W+1+\left( 1-\varphi \right) \left( \Delta -\pi \right) \right] .$

Note that $e^{N*}=\frac{1}{2}+\frac{1}{2}(W+\Delta -\pi )$ can be written as $\frac{1}{2}+\frac{1}{2}W=e^{N*}+\frac{1}{2}\left( \pi -\Delta \right)$.

Then we can also write

$$\begin{aligned} e^{S*}&=\frac{1}{2}\left[ W+1+\left( 1-\varphi \right) \left( \Delta -\pi \right) \right] \\ &=\frac{1}{2}+\frac{1}{2}W+\frac{1}{2}\left( 1-\varphi \right) \left( \Delta -\pi \right) \\&=e^{N*}+\frac{1}{2}\left( \pi -\Delta \right) +\frac{1}{2}\left( 1-\varphi \right) \left( \Delta -\pi \right) \\& =e^{N*}+\frac{1}{2}\left( \pi -\Delta \right) \left( 1-\frac{1}{2}\left( 1-\varphi \right) \right) \\&= e^{N*}+\frac{1}{4}\left( \pi -\Delta \right) \left( 1+\varphi \right) .\\ \end{aligned}$$

The last expression is larger than or equal to $e^{N*}$ if and only if $\frac{1}{4}\left( \pi -\Delta \right) \left( 1+\varphi \right) \ge 0$, or $\pi \ge \Delta$.

Lemma 3

Assume $\Delta >\pi$ and interior solutions of e.

Under no-GSP:

$$\begin{aligned} (1-e^{N*})D^{N*}=\left[ 1-\frac{1}{2}(1+W+\Delta -\pi )\right] \left[ \frac{1}{2}(\pi +W+\Delta -1)\right] \\ =\left[ \frac{1}{2}-\frac{1}{2}(W+\Delta -\pi )\right] \left[ \frac{1}{2}- \frac{1}{2}(W+\pi +\Delta )\right] (-1)>0\\ \end{aligned}$$

The condition is positive so first term is positive and second term is negative because $(W+\pi +\Delta )>(W+\Delta -\pi )$.

Under GSP:

$$\begin{aligned} (1-e^{S*})D^{S*}=\left[ \frac{1}{2}\left[ W+1+\left( 1-\varphi \right) \left( \Delta -\pi \right) \right] \right] \left[ \frac{1}{2}\left[ \left( 1-\varphi \right) \left( \pi +\Delta \right) +W-1\right] \right] \\ =\left[ \frac{1}{2}-\frac{1}{2}(W+\left( 1-\varphi \right) \left( \Delta -\pi \right) )\right] \left[ \frac{1}{2}-\frac{1}{2}(W+\left( 1-\varphi \right) \left( \Delta +\pi \right) )\right] (-1)>0\\ \end{aligned}$$

As above, the condition is positive so most likely first term is positive and second term is negative because

$$\begin{aligned} (W+\left( 1-\varphi \right) \left( \Delta +\pi \right) )>(W+\left( 1-\varphi \right) \left( \Delta -\pi \right) ).\\ \end{aligned}$$

Then, cross comparing terms under no-GSP and under GSP:

$$\begin{aligned} \left[ \frac{1}{2}-\frac{1}{2}(W+\left( \Delta -\pi \right) )\right] <\left[ \frac{1}{2}-\frac{1}{2}(W+\left( 1-\varphi \right) \left( \Delta -\pi \right) )\right] \end{aligned}$$

and

$$\begin{aligned} \left[ \frac{1}{2}-\frac{1}{2}(W+\pi +\Delta )\right] (-1)>\left[ \frac{ 1}{2}-\frac{1}{2}(W+\left( 1-\varphi \right) \left( \Delta +\pi \right) ) \right] (-1). \end{aligned}$$

The (increasing) variation in the first component is equal to $\frac{1}{2} \left( \Delta -\pi \right) \varphi$

The (decreasing) variation in the second component is equal to $\frac{1}{2} \left( \Delta +\pi \right) \varphi$

The stronger effect is the decreasing effect.

Hence, $(1-e^{N*})D^{N*}>(1-e^{S*})D^{S*}.$

Proposition 6

$$\begin{aligned} \begin{array}{l} \Pr ^{N}\left( G\right) =e^{*N}=W+\Delta -D^{*N}\\ \Pr ^{S}\left( G\right) =(1-\varphi )e^{*S}+\varphi =(1-\varphi )(W-D^{*S}+(1-\varphi )\Delta )+\varphi \\ \end{array} \end{aligned}$$

When $\Pr ^{N}\left( G\right) >\Pr ^{S}\left( G\right)$

$$\begin{aligned} \Pr ^{S}\left( G\right)>P(GG)+P(BG), then\, e^{*N}>e^{*S}+\varphi (1-e^{*S}) \end{aligned}$$

We know $e^{S*}=\frac{1}{2}\left[ W+1+\left( 1-\varphi \right) \left( \Delta -\pi \right) \right]$ and $e^{N*}=\frac{1}{2}(1+W+\Delta -\pi ),$ so

$$\begin{aligned} \begin{array}{l} e^{*N}>e^{*S}+\varphi (1-e^{*S})\Leftrightarrow e^{*N}>e^{*S}(1-\varphi )+\varphi \\ \Leftrightarrow \frac{1}{2}(1+W+\Delta -\pi )>\frac{1}{2}\left[ W+1+\left( 1-\varphi \right) \left( \Delta -\pi \right) \right] (1-\varphi )+\varphi \\ \Leftrightarrow 1+W+\Delta -\pi>[(W+1+\Delta -\pi )-\varphi \left( \Delta -\pi \right) ](1-\varphi )+2\varphi \\ \Leftrightarrow 1+W+\Delta -\pi>(W+1+\Delta -\pi )-\varphi \left( \Delta -\pi \right) -\varphi [(W+1+\Delta -\pi )\\ -\varphi \left( \Delta -\pi \right) ]+2\varphi \\ \Leftrightarrow 0>-\left( \Delta -\pi \right) -[(W+1+\Delta -\pi )-\varphi \left( \Delta -\pi \right) ]+2 \\ \Leftrightarrow 0>-\Delta +\pi -W-\Delta +\pi +\varphi \left( \Delta -\pi \right) +1\\ \Leftrightarrow 0>(2-\varphi )(\pi -\Delta )-W+1 \\ \Leftrightarrow W>(2-\varphi )(\pi -\Delta )+1\\ \end{array} \end{aligned}$$

Let $W_{\min }=(2-\varphi )(\pi -\Delta )+1.$

Taking into account the conditions stated in Lemma 2, it is easily proven that $W_{min}$ cannot lead to interior solutions.

Recall that $min\{E;F\}=F=\left( 1-\varphi \right) \left( \pi -\Delta \right) +1.$

It is always verified that $W_{\min }>F$. Thus, $W_{min}$ doesn’t satisfy the conditions of Lemma 2 and so $\Pr ^{N}\left( G\right) >\Pr ^{S}\left( G\right)$ cannot happen for the optimal interior solutions of e and D.