Reference pricing in the presence of pseudo-generics

Abstract

This paper looks at producers of branded and generic pharmaceuticals’ pricing decisions under two possible reimbursement schemes—reference pricing and fixed percentage reimbursement—and under two settings—one where the branded producer only sells the (off-patent) branded pharmaceutical and another where, in addition, it may also sell its own generic version, a so called pseudo-generic. We find different pricing responses from firms under the two reimbursement schemes and across the two settings analysed (with or without a pseudo-generic), and show that pseudo-generics may have an anticompetitive effect. Our results have important policy implications such as showing that the presence of pseudo-generics reinforces reference pricing’s advantages over alternative reimbursement schemes.

Appendix: Detailed pricing calculations

The market with no pseudo-generics

Fixed percentage reimbursement

Under a FPR scheme, the “marginal consumer”, $c_{b.g}^{F}$, who is indifferent between buying the branded $(b)$ or generic $(g)$ pharmaceuticals is found by solving $\gamma -\theta p_{g}^{F}-t\times c_{b.g}^{F}=\beta -\theta p_{b}^{F}$:

$$\begin{aligned} c_{b.g}^{F}=\frac{\theta (p_{b}^{F}-p_{g}^{F})-(\beta -\gamma )}{t} \end{aligned}$$

(23)

Let $\mathbf {p}^{F}=\left( p_{b}^{F},p_{g}^{F}\right)$ represent the headline price vector. As the total number of consumers is assumed to be equal to 1, the demand functions are given by:

$$\begin{aligned} B\left( \mathbf {p}^{F}\right)&= 1-c_{b.g}^{F}=1-\frac{\theta (p_{b}^{F}-p_{g}^{F})-(\beta -\gamma )}{t} \nonumber \\ G\left( \mathbf {p}^{F}\right)&= c_{b.g}^{F}=\frac{\theta (p_{b}^{F}-p_{g}^{F})-(\beta -\gamma )}{t} \end{aligned}$$

(24)

The profit functions are given by $\Pi _{BP}\left( \mathbf {p}^{F}\right) =p_{b}^{F}\times B\left( \mathbf {p}^{F}\right)$ and $\Pi _{GP}\left( \mathbf {p}^{F} \right) =p_{g}^{F}\times G\left( \mathbf {p}^{F}\right)$. Profit maximization with respect to $p_{b}^{F}$ and $p_{g}^{F}$ respectively yields the best-response functions:

$$\begin{aligned} p_{b}^{F}&= \frac{1}{2\theta }t+\frac{1}{2\theta }\left( \beta -\gamma \right) +\frac{1}{2}p_{g}^{F}\nonumber \\ p_{g}^{F}&= \frac{1}{2}p_{b}^{F}-\frac{1}{2\theta }\left( \beta -\gamma \right) \end{aligned}$$

(25)

In a Nash equilibrium, we obtain the following equilibrium prices:

$$\begin{aligned} p_{b}^{F}&= \frac{2}{3\theta }t+\frac{1}{3\theta }\left( \beta -\gamma \right) \nonumber \\ p_{g}^{F}&= \frac{1}{3\theta }t-\frac{1}{3\theta }\left( \beta -\gamma \right) \end{aligned}$$

(26)

Therefore, equilibrium quantities are given by:

$$\begin{aligned} B^{F}&= \frac{2}{3}+\frac{1}{3t}\left( \beta -\gamma \right) \nonumber \\ G^{F}&= \frac{1}{3}-\frac{1}{3t}\left( \beta -\gamma \right) \end{aligned}$$

(27)

and equilibrium profits are:

$$\begin{aligned} \Pi _{BP}^{F}&= \frac{1}{\theta }\left( \frac{4}{9}t+\frac{4}{9} \left( \beta -\gamma \right) +\frac{\left( \beta -\gamma \right) ^{2}}{9t}\right) \nonumber \\ \Pi _{GP}^{F}&= \frac{1}{\theta }\left( \frac{1}{9}t-\frac{2}{9} \left( \beta -\gamma \right) +\frac{\left( \beta -\gamma \right) ^{2}}{9t}\right) \end{aligned}$$

(28)

Consumer surplus is given by the sum of the surplus of buying the branded and generic pharmaceuticals (in equilibrium, $c_{b.g}^{F}=\frac{t+\gamma -\beta }{3t})$:

$$\begin{aligned} CS^{F}&= \int _{0}^{\frac{t+\gamma -\beta }{3t}}\left( \gamma -\theta p_{g}^{F}-t\times c\right) dc+\int _{\frac{t+\gamma -\beta }{3t}}^{1}\left( \beta -\theta p_{b}^{F}\right) dc\nonumber \\&= \frac{\left( \beta -\gamma \right) ^{2}-11t^{2}+10\beta t +8\gamma t}{18t} \end{aligned}$$

(29)

Total profits, which are equivalent to total pharmaceutical expenditure, are given by:

$$\begin{aligned} \Pi ^{F}&= \Pi _{BP}^{F}+\Pi _{GP}^{F} \nonumber \\&= \frac{1}{\theta }\left( \frac{5}{9}t+\frac{2}{9}\left( \beta -\gamma \right) +\frac{2}{9}\frac{\left( \beta -\gamma \right) ^{2}}{t} \right) \end{aligned}$$

(30)

Government (or other third-party payers) expenditure with pharmaceuticals is a proportion $\left( 1-\theta \right)$ of total pharmaceutical expenditure:

$$\begin{aligned} G^{F}&= \left( 1-\theta \right) \Pi ^{F} \nonumber \\&= \frac{1-\theta }{\theta }\left( \frac{5}{9}t+\frac{2}{9} \left( \beta -\gamma \right) +\frac{2}{9}\frac{\left( \beta - \gamma \right) ^{2}}{t}\right) \end{aligned}$$

(31)

Parameter restrictions    Two restrictions must hold for these results to be valid. Firstly, $0<c_{b.g}^{F}<1,$ i.e., both the branded and the generic varieties are sold in equilibrium. $c_{b.g}^{F}<1$ is always verified, whilst for $c_{b.g}^{F}>0$ to be verified, the following condition must hold:

$$\begin{aligned} \beta <t+\gamma \end{aligned}$$

(32)

Secondly, $\beta \ge \hat{p}_{b}^{F}$, i.e., CS must be positive (in equilibrium) for both product varieties. In order for this to be verified, the following must hold:

$$\begin{aligned} \beta \ge t-\frac{\gamma }{2} \end{aligned}$$

(33)

These restrictions are similar to those of Rodrigues et al. (2014).²²

Reference pricing

When a reference pricing scheme is in place, the consumer who is indifferent between purchasing a generic or a branded pharmaceutical is found by setting $\gamma -\hat{p}_{g}^{R}-t\times c_{b.g}^{R}=\beta -\hat{p}_{b}^{R},$ which yields:

$$\begin{aligned} c_{b.g}^{R}=\frac{(p_{b}^{R}-p_{g}^{R})-(\beta -\gamma )}{t} \end{aligned}$$

(34)

Note that $c_{b.g}^{R}=c_{b.g}^{NR}$, that is, the marginal consumer under reference pricing has the same location as the marginal consumer when no reimbursement scheme exists [found by setting $\theta =1$ in Eq. (23)]. Therefore, demand, profit functions, equilibrium quantities and prices are equal in these two scenarios: $p_{k}^{R}=p_{k}^{NR}$, with $k=b,g$.

Consumer surplus under a RP scheme is given by²³:

$$\begin{aligned} CS^{R}&= \int _{0}^{\frac{t+\gamma -\beta }{3t}}\left( \gamma -\theta p_{g}^{R}-t\times c\right) dc+\int _{\frac{t+\gamma -\beta }{3t}}^{1} \left( \beta -p_{b}^{R}+\left( 1-\theta \right) p_{g}^{R}\right) dc \nonumber \\&= \frac{\left( \beta -\gamma \right) ^{2}-\left( 5+6\theta \right) t^{2}+\left( 6\theta +4\right) \beta t+\left( 14-6\theta \right) \gamma t}{18t} \end{aligned}$$

(35)

Total profits are given by:

$$\begin{aligned} \Pi ^{R}&= \Pi _{BP}^{R}+\Pi _{GP}^{R} \nonumber \\&= \frac{5}{9}t+\frac{2}{9}\left( \beta -\gamma \right) + \frac{2}{9}\frac{\left( \beta -\gamma \right) ^{2}}{t} \end{aligned}$$

(36)

Government expenditure is a proportion $\left( 1-\theta \right)$ of the reference price (because the total number of consumers is equal to one):

$$\begin{aligned} G^{R}&= \left( 1-\theta \right) \left( B^{R}+G^{R}\right) p_{g}^{R}\nonumber \\&= \left( 1-\theta \right) p_{g}^{R} \nonumber \\&= \left( 1-\theta \right) \left( \frac{1}{3}t-\frac{1}{3} \left( \beta -\gamma \right) \right) \end{aligned}$$

(37)

Parameter restrictions    As outlined earlier, two conditions must be verified: $0<c_{b.g}^{R}<1$ and $\beta \ge \hat{p}_{b}^{R}$. In the first case, the same restriction outlined earlier must also hold: $\beta <t+\gamma$. As for the second case, the following must hold: $\beta \ge t-\gamma \frac{\left( 2- \theta \right) }{\left( 1+\theta \right) }$. This latter condition is less restrictive than the previous second restriction given by Eq. (33) for any $\theta \in \left[ 0,1\right]$; therefore, it is always satisfied when Eq. (33) holds.

The market with a pseudo-generic

Fixed percentage reimbursement

Under FPR and with a pseudo-generic in the market, we assume that it is located sufficiently close to the generic so that consumer $c_{g.pg}^{F}$ is indifferent between the two. Solving $\gamma -\hat{p}_{g}^{F}-tc_{g.pg}^{F}=\gamma -\hat{p}_{pg}^{F}-t \left( f_{BP}-c_{g.pg}^{F}\right)$ we find:

$$\begin{aligned} c_{g.pg}^{F}=\frac{f_{BP}}{2}+\frac{\theta \left( p_{pg}^{F}-p_{g}^{F}\right) }{2t} \end{aligned}$$

(38)

In addition, provided the pseudo-generic does not fully cannibalize the sales of the branded variety, consumer $c_{b.pg}^{r,F}$ will be indifferent between the two ($r$ indicates that this consumer is located to the right of $f_{BP}$). Solving $\gamma -\hat{p}_{pg}^{F}-t\left( c_{b.pg}^{r,F}-f_{BP}\right) =\beta -\hat{p}_{b}^{F}$ we find:

$$\begin{aligned} c_{b.pg}^{r,F}=f_{BP}-\frac{\beta -\gamma }{t}+\frac{\theta \left( p_{b}^{F}-p_{pg}^{F}\right) }{t} \end{aligned}$$

(39)

Let $\underline{\mathbf {p}}^{F}=\left( p_{b}^{F}, p_{pg}^{F}, p_{g}^{F}\right)$ be the headline price vector under a FPR scheme. Demand functions are thus given by:

$$\begin{aligned} B\left( \underline{\mathbf {p}}^{F}\right)&= 1-c_{b.pg}^{r,F}= 1-f_{BP}+\frac{\beta -\gamma }{t}-\frac{\theta \left( p_{b}^{F} -p_{pg}^{F}\right) }{t}\nonumber \\ PG\left( \underline{\mathbf {p}}^{F}\right)&= c_{b.pg}^{r,F}- c_{g.pg}^{F}=\frac{f_{BP}}{2}-\frac{\beta -\gamma }{t}+ \frac{\theta \left( p_{b}^{F}-p_{pg}^{F}\right) }{t}- \frac{\theta \left( p_{pg}^{F}-p_{g}^{F}\right) }{2t} \nonumber \\ G\left( \underline{\mathbf {p}}^{F}\right)&= c_{g.pg}^{F}= \frac{f_{BP}}{2}+\frac{\theta \left( p_{pg}^{F}-p_{g}^{F}\right) }{2t} \end{aligned}$$

(40)

In this scenario, the BP produces both the branded and the pseudo-generic products. Hence, its profit function is given by $\Pi _{BP}\left( \underline{\mathbf {p}}^{F}\right) =p_{b}^{F}\times B\left( \underline{\mathbf {p}}^{F}\right) +p_{pg}^{F}\times PG \left( \underline{\mathbf {p}}^{F}\right)$, whilst the generic producer’s profit function is given by $\Pi _{GP}\left( \underline{\mathbf {p}}^{F}\right) =p_{g}^{F}\times G\left( \underline{\mathbf {p}}^{F}\right)$. Maximizing the former with respect to $p_{b}^{F}$ and $p_{pg}^{F}$ and the latter with respect to $p_{g}^{F}$, we find the best-response functions which lead to the following Nash equilibrium prices:

$$\begin{aligned} \underline{p}_{b}^{F}&= \frac{11}{6\theta }t-\frac{5}{6\theta } tf_{BP}+\frac{1}{2\theta }\left( \beta -\gamma \right) \nonumber \\ \underline{p}_{pg}^{F}&= \frac{4}{3\theta }t-\frac{1}{3\theta } tf_{BP}\nonumber \\ \underline{p}_{g}^{F}&= \frac{2}{3\theta }t+\frac{1}{3\theta } tf_{BP} \end{aligned}$$

(41)

At these prices, we obtain the equilibrium quantities:

$$\begin{aligned} \underline{B}^{F}&= \frac{1}{2}\left( 1-f_{BP}\right) +\frac{\beta -\gamma }{2t}\nonumber \\ \underline{PG}^{F}&= \frac{1}{6}\left( 1+2f_{BP}\right) -\frac{\beta -\gamma }{2t} \nonumber \\ \underline{G}^{F}&= \frac{1}{3}+\frac{1}{6}f_{BP} \end{aligned}$$

(42)

And the equilibrium profits are:

$$\begin{aligned} \underline{\Pi }_{BP}^{F}&= \frac{1}{36\theta }\left( t \left( 41-f_{BP}\left( 34-11f_{BP}\right) \right) +18\left( 1-f_{BP}\right) \left( \beta -\gamma \right) +\frac{9\left( \beta -\gamma \right) ^{2}}{t}\right) \nonumber \\ \underline{\Pi }_{GP}^{F}&= \frac{1}{18\theta }t\left( 2+ f_{BP}\right) ^{2} \end{aligned}$$

(43)

Consumer surplus is given by²⁴:

$$\begin{aligned} \underline{CS}^{F}&= \int _{0}^{\frac{f_{BP}}{6}+\frac{1}{3}} \left( \gamma -\theta \underline{p}_{g}^{F}-t\times c\right) dc+ \int _{\frac{f_{BP}}{6}+\frac{1}{3}}^{f_{BP}}\left( \gamma - \theta \underline{p}_{pg}^{F}-t\times \left( f_{BP}-c\right) \right) dc \nonumber \\&\quad +\int _{f_{BP}}^{\frac{\left( 1+f_{BP}\right) t-\left( \beta -\gamma \right) }{2t}}\left( \gamma -\theta \underline{p}_{pg}^{F}- t\times \left( c-f_{BP}\right) \right) dc+\int _{\frac{\left( 1+ f_{BP}\right) t-\left( \beta -\gamma \right) }{2t}}^{1}\left( \beta -\theta \underline{p}_{b}^{F}\right) dc \nonumber \\&= \frac{9\left( \beta -\gamma \right) ^{2}+\left( 18\beta +54\gamma \right) t-18\left( \beta -\gamma \right) tf_{BP}-\left( 115-86f_{BP}+ 61f_{BP}^{2}\right) t^{2}}{72t} \end{aligned}$$

(44)

Total profits are given by:

$$\begin{aligned} \underline{\Pi }^{F}&= \underline{\Pi }_{BP}^{F}+ \underline{\Pi }_{GP}^{F}\nonumber \\&= \frac{1}{36\theta }\left( t\left( 49-f_{BP}\left( 26-13f_{BP} \right) \right) +18\left( 1-f_{BP}\right) \left( \beta -\gamma \right) +\frac{9\left( \beta -\gamma \right) ^{2}}{t}\right) \end{aligned}$$

(45)

Government expenditure is given by:

$$\begin{aligned} \underline{G}^{F}&= \left( 1-\theta \right) \underline{\Pi }^{F}\nonumber \\&= \frac{\left( 1-\theta \right) }{36\theta }\left( t\left( 49-f_{BP} \left( 26-13f_{BP}\right) \right) +18\left( 1-f_{BP}\right) \left( \beta -\gamma \right) +\frac{9\left( \beta -\gamma \right) ^{2}}{t}\right) \end{aligned}$$

(46)

Parameter restrictions    Five conditions must hold for these results to be valid. Firstly, $0<c_{b.g}^{r,F}<1,$ which is equivalent to requiring that:

$$\begin{aligned} \gamma +\left( \frac{1}{3}-\frac{7}{3}f_{BP}\right) t<\beta <\gamma +\frac{7}{3}\left( 1-f_{BP}\right) t \end{aligned}$$

(47)

Secondly, $c_{b.pg}^{r,F}>c_{b.g}^{F}$ and $\gamma -\underline{\hat{p}}_{pg}^{F}-tf_{BP}<\gamma -\underline{\hat{p}}_{g}^{F}$, i.e., the pseudo-generic is sold in equilibrium. The latter is satisfied provided $\underline{\hat{p}}_{pg}^{F}>\underline{\hat{p}}_{g}^{F}$, which holds in equilibrium. The former is satisfied provided:

$$\begin{aligned} f_{BP}>2/5 \end{aligned}$$

(48)

If this condition holds, the first inequality in Eq. (47) is always satisfied for any $\beta >\gamma$.

Thirdly, $f_{BP}<c_{b.pg}^{r,F}<1,$ so that the branded product is sold in equilibrium. $c_{b.pg}^{r,F}<1$ is always satisfied, but in order for $f_{BP}<c_{b.pg}^{r,F}$ the following must hold:

$$\begin{aligned} \beta <\gamma +\left( 1-f_{BP}\right) t \end{aligned}$$

(49)

Fourthly, $c_{b.g}^{F}>c_{b.pg}^{l,F}$ so that the pseudo-generic is sold in equilibrium. This is equivalent to requiring that:

$$\begin{aligned} \beta <\gamma +\left( \frac{5}{3}-\frac{8}{3}f_{BP}\right) t \end{aligned}$$

(50)

This condition is more restrictive than the second inequality in Eq. (47) and the inequality in Eq. (49). In addition, this condition sets an upper boundary for $f_{BP},$ because by assumption $\beta >\gamma$. Hence, for this assumption to hold, $f_{BP}<5/8$. Together with Eq. (48), this implies that:

$$\begin{aligned} 2/5<f_{BP}<5/8 \end{aligned}$$

(51)

Finally, all product varieties must provide positive surplus, which is equivalent to requiring that $\beta \ge \underline{\hat{p}}_{b}^{F}$. This is verified provided the following condition holds:

$$\begin{aligned} \beta \ge \frac{t\left( 11-5f_{BP}\right) }{3}-\gamma \end{aligned}$$

(52)

Therefore, Eqs. (50), (51) and (52) must hold for our results to be verified.²⁵

Reference pricing

Under RP, the location of the consumer who is indifferent between the branded and pseudo-generic products and between the pseudo-generic and the generic products is equivalent to the NR scenario: solving $\gamma -\hat{p}_{g}^{R}-tc_{g.pg}^{R}=\gamma -\hat{p}_{pg}^{R}-t\left( f_{BP}-c_{g.pg}^{R}\right)$ we obtain the latter $(c_{g.pg}^{R}=\frac{f_{BP}}{2}+\frac{p_{pg}^{R}- p_{g}^{R}}{2t})$, whilst solving $\gamma -\hat{p}_{pg}^{R} -t\left( c_{b.pg}^{r,R}-f_{BP}\right) =\beta -\hat{p}_{b}^{R}$ we find the former $(c_{b.pg}^{r,R}=f_{BP}-\frac{\beta -\gamma }{t} +\frac{p_{b}^{R}-p_{pg}^{R}}{t})$. Hence, equilibrium quantities, prices and profits are equal to those in that scenario [in particular, $\underline{p}_{k}^{R}=\underline{p}_{k}^{NR}$, with $k=b,pg,g$, and the latter can be found by setting $\theta =1$ in Eq. (41)].

Consumer surplus is given by²⁶:

$$\begin{aligned} \underline{CS}^{R}&= \int _{0}^{\frac{f_{BP}}{6}+\frac{1}{3}} \left( \gamma -\theta \underline{p}_{g}^{R}-t\times c\right) dc+ \int _{\frac{f_{BP}}{6}+\frac{1}{3}}^{f_{BP}}\left( \gamma - \underline{p}_{pg}^{R}+\left( 1-\theta \right) \underline{p}_{g}^{R} -t\times \left( f_{BP}-c\right) \right) dc\nonumber \\&\quad +\int _{f_{BP}}^{\frac{\left( 1+f_{BP}\right) t-\left( \beta -\gamma \right) }{2t}}\left( \gamma -\underline{p}_{pg}^{R}+ \left( 1-\theta \right) \underline{p}_{g}^{R}-t\times \left( c-f_{BP}\right) \right) dc \nonumber \\&\quad +\int _{\frac{\left( 1+f_{BP}\right) t-\left( \beta - \gamma \right) }{2t}}^{1}\left( \beta -\underline{p}_{b}^{R}+ \left( 1-\theta \right) \underline{p}_{g}^{R}\right) dc \nonumber \\&= \frac{9\left( \beta -\gamma \right) ^{2}+\left( 18\beta +54\gamma \right) t-18\left( \beta -\gamma \right) tf_{BP}- \left( 67+48\theta -110f_{BP}+24\theta f_{BP}+61f_{BP}^{2}\right) t^{2}}{72t} \end{aligned}$$

(53)

Total profits are given by:

$$\begin{aligned} \underline{\Pi }^{R}&= \underline{\Pi }_{BP}^{R}+ \underline{\Pi }_{GP}^{R}\nonumber \\&= \frac{1}{36}\left( t\left( 49-f_{BP}\left( 26-13f_{BP}\right) \right) +18\left( 1-f_{BP}\right) \left( \beta -\gamma \right) +\frac{9 \left( \beta -\gamma \right) ^{2}}{t}\right) \end{aligned}$$

(54)

Recall that total profits under a RP scheme are equivalent to total profits under a NR scheme, i.e., $\underline{\Pi }^{R}=\underline{\Pi }^{NR}$.

Government expenditure is given by:

$$\begin{aligned} \underline{G}^{R}&= \left( 1-\theta \right) \left( \underline{B}^{R} +\underline{PG}^{R}+\underline{G}^{R}\right) \underline{p}_{g}^{R}\nonumber \\&= \left( 1-\theta \right) \underline{p}_{g}^{R} \nonumber \\&= \left( 1-\theta \right) \left( \frac{2}{3}t+\frac{1}{3}tf_{BP}\right) \end{aligned}$$

(55)

Parameter restrictions    As in the FPR scheme case, Eqs. (50), (51) must hold for our results to be verified. In addition, $\beta \ge \underline{\hat{p}}_{b}^{R},$ which is equivalent to requiring that $\beta \ge \frac{\left[ \left( 7+4\theta \right) -\left( 7- 2\theta \right) f_{BP}\right] }{3}-\gamma$. This condition is less restrictive than Eq. (52) for any $\theta \in \left[ 0,1\right]$; therefore, it is always satisfied when Eq. (52) holds.