The effects of rebate contracts on the health care system

Abstract

Group purchasing organizations gain increasing importance with respect to the supply of pharmaceutical products and frequently use multiple, exclusive or partially exclusive rebate contracts to exercise market power. Based on a Hotelling model of horizontal and vertical product differentiation, we examine the controversy around whether a superior rebate scheme exists, as far as consumer surplus, firms’ profits and total welfare are concerned. We find that firms clearly prefer partially exclusive over multiple, and multiple over exclusive rebate contracts. In contrast, no rebate form exists that lowers total costs per se for the consumers or maximizes total welfare.

Appendix

Proof

Multiple Rebate Contracts: For the indifferent consumer

$$ V-p_1-tx+\frac{rx^2}{x} = V-p_2-t(1-x)+\frac{r(1-x)^2}{(1-x)} -\beta $$

has to hold, which yields the position of the indifferent consumer at $\frac{p_2-p_1+t+\beta-r}{2t-2r}.$ Firm i’s maximization problem is given by

$$ \max_{\substack{p_i}} = (p_i-c)D_i(p_i, p_j)-rD_i(p_i, p_j))^2. $$

The first order condition is

$$ \frac{\partial \pi_i^{{\rm MR}}}{\partial p_i} = \frac{p_i (r-2 t)+c (-r+t)+t (p_j-r+t+\beta )}{2 (r-t)^2} = 0. $$

This yields the solutions of $p_2^{{\rm MR}} = c+t-\frac{t \beta }{3 t-r}$ and $c+t+\frac{t \beta }{3 t-r} = p_1^{{\rm MR}}$.