Cocktails done right: price competition and welfare when substitutes become complements

Abstract

In this paper we analyze the effects of the introduction (by either firms or authorities) of a composite good consisting of a fixed proportion of two imperfectly substitutable stand-alone products. First, we find that such a “cocktail” rises the Bertrand equilibrium prices as it introduces a certain degree of complementarity. It also creates incentives to price discriminate and products can be sold at a discount or at a premium (depending on their degree of substitutability) when they are used as part of the composite good. We consider two distinct forms of price discrimination: a traditional one, in which producers set their prices independently of each other and a coordinated one, in which producers cooperate (collude) when setting the price of the composite good. Composite goods might have either a positive or a negative impact on consumer surplus. The sign of the impact depends on the form of price discrimination and consumers tend to be better off if producers coordinate. The impact is also more likely to be positive if “cocktails are done right”, i.e., if their quality is high compared to the quality of the stand-alone products.

Appendix

Proof of Proposition 1

From straightforward profit maximization, we have the following:

• If$\alpha _\mathbf {3}=\mathbf {1}$, the monopolist chooses in equilibrium the same prices it would choose without a cocktail, namely $p_{Mi}^{*}=\frac{1}{2}$. Each treatment serves a demand equal to $q_{M1}^*=q_{M2}^*=q_{M3}^*=\frac{1}{6}$. Thus, the total quantity sold of good i is $q_i^*=\frac{1}{6}+r_i\,\frac{1}{6}=\frac{1}{4}$, $i=1,2$, whereas the total number of treated patients is $q_{M1}^*+q_{M2}^*+q_{M3}^*=\frac{1}{2}$, $\frac{1}{3}$ of it is cured with the cocktail. Quantities thus are $q_{Mi}^{*}=\frac{1}{4}$, $i=1,2$, while profits equal $\varPi _M^*=\frac{1}{4}$.

• If$\alpha _\mathbf {3}>\mathbf {1}$, according to Eq. (13), $q_1$ and $q_2$ are sold both as stand-alone drugs and in a cocktail if $\alpha _3<\alpha _{3}(\gamma )$. Define $\alpha _{3i}^{*M}$ the value taken by $\alpha _{3}(\gamma )$ in this case, where the superscript M stands for “monopolist”. We shall determine the exact value of $\alpha _{3i}^{*M}$ below. Assume first that $\alpha _3<\alpha _{3i}^{*M}$.

  The two prices are set at $p_{M1}^{\alpha<\alpha _{3i}^{*}}=p_{M2}^{\alpha <\alpha _{3i}^{*}}=\frac{2+\alpha _{3}}{6}$ (where $p_{Mi}^{\alpha <\alpha _{3i}^{*}}>p_{Mi}^{*}$: the prices the monopolist charges are higher with the cocktail). Monopoly profits are $\varPi _M^{\alpha <\alpha _{3}^{*M}}=\frac{(2+\alpha _3)^2}{36}$, which are always greater than $\varPi _M^*$, the profits obtained in the absence of the cocktail.

  Substituting $p_{M1}^{\alpha <\alpha _{3i}^{*}}$ and $p_{M2}^{\alpha <\alpha _{3i}^{*}}$ into either the expression for $q_1$ in (9) or for $q_2$ in (10)

  $$\begin{aligned} \alpha _{3}(\gamma )=\frac{2(2-\gamma )}{(1+\gamma )}=\alpha _{3i}^{*M} \end{aligned}$$

  (22)

  Comparing $\alpha _{3i}^{*M}$ with $\alpha _3^*(\gamma )$ in Eq. (15), $\alpha _{3i}^{*M}\ge \alpha _3^*(\gamma )$ if $\gamma \le \frac{1}{3}$. In such a case, our restriction $\alpha _3\in [1,\alpha _3^*(\gamma ))$ ensures that $q_1$ and $q_2$ are always sold both as stand-alone drug and in the cocktail.

  If $\gamma \ge \frac{1}{3}$ and $\alpha _{3} \in [\alpha _{3i}^{M}, \alpha _{3}^{*}(\gamma )]$, we have a corner solution where $q_1=q_2=0$ and $q_3$ is given by (14).

  Maximizing profits in such case, the equilibrium cocktail price is $p_{M3}^{\alpha _3 > \alpha _{3i}^{*M}}=\frac{\alpha _3}{2}$, so that $q_{M3}^{\alpha _3 > \alpha _{3i}^{*M}}=\frac{\alpha _3}{2(3-2\gamma )}$ and $\varPi _M^{\alpha _3 > \alpha _{3i}^{*M}}=\frac{\alpha _3^2}{4(3-2\gamma )}$. Prices are higher than in the case with $\alpha _{\mathbf {3}}<\alpha _{\mathbf {3i}}^{*M}$ and without a cocktail. Profits are also higher.

  We now consider the impact of a cocktail on consumer surplus. Consumer surplus is defined as

$$\begin{aligned} CS =U(q_{1},q_{2},q_{3},M)-\left\{ \sum _{i=1}^3 p_i q_i-M\right\} \end{aligned}$$

(23)

Substituting equilibrium quantities and prices, in a multi-product monopoly, consumer surplus when the cocktail is not sold is

$$\begin{aligned} CS _M^{nc}=\frac{1}{8}. \end{aligned}$$

(24)

When $\alpha _3=1$, the introduction of the cocktail has no effect on consumer surplus, so that

$$\begin{aligned} CS _M^{\alpha _3=1}=\frac{1}{8}. \end{aligned}$$

(25)

Both when $\gamma <\frac{1}{3}$ and when $\gamma \ge \frac{1}{3}$ but $\alpha _{3} \in (1, \alpha _{3i}^{*M})$ consumer surplus is

$$\begin{aligned} CS _M^{\alpha _3<\alpha _{3i}^{*M}}=\frac{\alpha _3^2 (9-\gamma )-4 \alpha _3 (3+\gamma )+4 (3-\gamma )}{72 (1-\gamma )} \end{aligned}$$

(26)

which is always greater than $\frac{1}{8}$.

Finally, when $\gamma \ge \frac{1}{3}$ and $\alpha _{3} \in [\alpha _{3}^{*M}, \alpha _{3}^{*}(\gamma )]$, consumer surplus is

$$\begin{aligned} CS _M^{\alpha _3>\alpha _{3}^{*M}}=\frac{\alpha _3^2}{8(3-2\gamma )} \end{aligned}$$

(27)

Notice that in this case $CS _M^{\alpha _3>\alpha _{3}^{*M}}>\frac{1}{8}$, as well. Quite intuitively, when $\alpha _3$ is high, consumers benefit from the presence of the cocktail. $\square$

Proof of Proposition 2

Without the cocktail, the two firms charge prices $p_i^{nc}=\frac{2(1-\gamma )}{4-3\gamma }$, $i=1,2$, while the demands of the two drugs are $q_{i}^{nc}=\frac{(2-\gamma )}{8-6\gamma }$, $i=1,2$. This yields profits $\varPi _i^{nc}=\frac{(2-\gamma )(1-\gamma )}{(4-3\gamma )^2}$. Using the general expression in (23), consumer surplus in this case

$$\begin{aligned} CS ^{nc}=\frac{(2-\gamma )^2}{2(4-3\gamma )^2}. \end{aligned}$$

(28)

With the cocktail, the equilibrium prices are $p_{1}^{c}=p_{2}^{c}=\frac{6(1-\gamma )}{11-9\gamma }$, while the quantities sold of each regimen (i.e., the demands of the two separate drugs, $q_{1}^{c}$, $q_{2}^{c}$ and that of the cocktail $q_{3}^{c}$) are $q_{i}^{c}= \frac{(5-3\gamma )}{33-27\gamma }$, $i=1,2,3$. Profits are $\varPi _i^c=\frac{3(1-\gamma )(5-3\gamma )}{(11-9\gamma )^2}$, $i=1,2$. Consumer surplus is

$$\begin{aligned} CS ^c_{\alpha _3=1}=\frac{(5-3 \gamma )^2}{2 (11-9 \gamma )^2} \end{aligned}$$

(29)

Comparison between prices, quantities, profits and consumer surplus yields the results.³²$\square$

Proof of Proposition 3

Assume that $\alpha _3 \in (1,\alpha _{3}^{*}(\gamma ))$, so that all demands in (9)–(11) are positive.

• a. In such case, the Bertrand equilibrium prices are: $p_{i}^{c}=\frac{2(2+\alpha _{3})(1-\gamma )}{11-9\gamma },\; i=1,2$. The equilibrium quantities of the two goods, including both the quantities consumed as stand -alone products and that included in cocktails, is $q_{i}^{c}=\frac{\alpha _3 \left( 3 \gamma ^2+\gamma -6\right) +6 \gamma ^2-25 \gamma +21}{9 \left( 9 \gamma ^2-20 \gamma +11\right) },\; i=1,2$. The quantity of the cocktail consumed in equilibrium is $q_3^c=\frac{\alpha _3 \left( 3 \gamma ^2-26 \gamma +27\right) +2 \left( 3 \gamma ^2+\gamma -6\right) }{9 (\gamma -1) (9 \gamma -11)}$, which is greater than $q_{i}^{c},\; i=1,2$, since the prices of the three products are the same and $\alpha _3>1$. The equilibrium profits are $\varPi _{i}^{c}=\frac{(2+\alpha _{3})^{2}(1-\gamma )(5-3\gamma )}{3(11-9\gamma )^{2}},$$i=1,2$. It is straightforward to check that $p_{\grave{\imath }}^{c}>p_{i}^{nc}$ and $q_{\grave{\imath }}^{c}<q_{i}^{nc}$, for any $\gamma \in [0,1)$.

• b. Profits $\varPi _{i}^{c}$ are increasing in $\alpha _{3}$, whereas $\varPi _{i}^{nc}$ is obviously invariant with respect to $\alpha _3$. It is immediate to show that $\varPi _{i}^{c}>\varPi _{i}^{nc}$ if $\alpha _3=1$ and $\gamma \ge 0.175$. Being $\varPi _i^{c}$ increasing in $\alpha _3$, this means that, for $\gamma \ge 0.175$, $\varPi _{i}^{c}\ge \varPi _{i}^{nc}$$\forall \alpha _3 \in (1,\alpha _{3}(\gamma ))$.

  If $\gamma < 0.175$, $\varPi _{i}^{c}< \varPi _{i}^{nc}$ at $\alpha _3=1$, whereas $\varPi _{i}^{c}> \varPi _{i}^{nc}$ at $\alpha _3= \alpha _{3}(\gamma )$. Hence, there exists ${\bar{\alpha }}_3 \in (1,\alpha _{3}^{*}(\gamma ))$ such that $\varPi _{i}^{c}\gtrless \varPi _{i}^{nc}$ if $\alpha _3\gtrless {\bar{\alpha }}_3$.

• c. We turn now to consumer surplus, again defined by Eq. (23). Substituting prices and quantities, consumer surplus in a duopoly without a cocktail is given by Eq. (28), whereas consumer surplus with the cocktail is

  $$\begin{aligned} CS _d^{\alpha _3}& {}= \frac{\alpha _3^2 \left( 9 \gamma ^3-201 \gamma ^2+451 \gamma -267\right) +4 \alpha _3 \left( 9 \gamma ^3+42 \gamma ^2-143 \gamma +96\right) }{18 (11-9 \gamma )^2 (\gamma -1)}\nonumber \\ & \quad +\,\frac{36 \gamma ^3-318 \gamma ^2+616 \gamma -342}{18 (11-9 \gamma )^2 (\gamma -1)}\ \end{aligned}$$

  (30)

  Given $\alpha _3 \in (1,\alpha _{3}(\gamma ))$, there exists $1<{\tilde{\alpha }}_3<\alpha _{3}(\gamma )$ such that $CS _d^{\alpha _3}\gtrless CS _d^{nc}$ if $\alpha _3\gtrless {\tilde{\alpha }}_3$.

  In fact, $CS _d^{\alpha _3}$ is increasing in $\alpha _3$ and $CS _d^{nc}$ does not depend on $\alpha _3$. $CS _d^{\alpha _3}< CS _d^{nc}$ at $\alpha _3=1$, whereas $CS _d^{\alpha _3}> CS _d^{nc}$ at $\alpha _3=\alpha _{3i}^{*D}$.

  It is worth mentioning that numerical simulations indicate that, when $\gamma <0.175$, ${\tilde{\alpha }}_3>{\bar{\alpha }}_3$. The cutoff ${\tilde{\alpha }}_3$ is decreasing in $\gamma$.

  Finally, at the equilibrium prices

  $$\begin{aligned} \alpha _3(\gamma )=\frac{6\gamma ^2-25\gamma +21}{6-\gamma -3\gamma ^2}=\alpha _{3i}^{*D} \end{aligned}$$

  (31)

  which is exactly the value chosen for our cutoff $\alpha _3^*(\gamma )$ in Eq. (15). We will show in Sect. 7.2  that $\alpha _{3i}^{*D}$ is the value that guarantees that the standalone treatments are always sold in duopoly.

Proof of Proposition 4

If the monopolist price discriminates, it maximizes profits with respect to $p_1$, $p_2$ and $p_3$. Substituting the expressions for demands $q_1$, $q_2$ and $q_3$ from (9), (10) and (11), equilibrium prices are $p_{Mi}^d=\frac{1}{2}$ ($i=1,2$, where the superscript d stands for “discrimination ”), and $p_{M3}^d=\frac{\alpha _3}{2}$. Equilibrium quantities are $q_{Mi}^d=\frac{3-(2+\alpha _3)\gamma }{18(1-\gamma )}$, $i=1,2$ and $q_{M3}^d=\frac{(3-\gamma )\alpha _3-2\gamma }{18(1-\gamma )}$. Equilibrium profits are $\varPi _{M}^{d}=\frac{6+(3-\gamma )\alpha _3^2 -4\gamma (1+\alpha _3)}{36(1-\gamma )}$.

Notice first that, given the equilibrium prices, $q_1$ and $q_2$ are positive iff

$$\begin{aligned} \alpha _3<\alpha _{3i}^{*Md}=\frac{3-2\gamma }{\gamma } \end{aligned}$$

(32)

However, it is easy to check that $\alpha _{3i}^{*Md}>\alpha _{3}(\gamma )$. Differently from uniform pricing, then, the possibility to price discriminate always implies that goods 1 and 2 will be sold both as stand-alone treatments and as parts of the cocktail. In such parameter’s range the monopolist charges a premium on the goods sold in the cocktail, since $p_{M3}^{d}=\frac{\alpha _3}{2}>r_1 p_{M1}^{d}+r_2 p_{M2}^{d}=\frac{1}{2}$. Particularly, the premium is equal to $\delta _M=p_{M3}^{d}-\frac{1}{2}=\frac{\alpha _3-1}{2}$.

Finally, it can be proven both that $\varPi _{M}^{d}>\varPi _M^{\alpha _3<\alpha _{3}^{*M}}$ and that $\varPi _{M}^{d}>\varPi _M^{\alpha _3>\alpha _{3}^{*M}}$. Thus, the monopolist always finds it profitable to price discriminate.

Using expression (23), consumer surplus with price discrimination when $\alpha _3<\alpha _{3}(\gamma )$ and goods are sold both as stand alone and in the cocktail is

$$\begin{aligned} CS _M^d=\frac{\alpha _3^2 (3-\gamma )-4 \alpha _3 \gamma +2(3-2 \gamma )}{72 (1 -\gamma )}. \end{aligned}$$

(33)

Under uniform pricing, both when $\gamma <\frac{1}{3}$ and when $\gamma \ge \frac{1}{3}$ but $\alpha _{3} \in (1, \alpha _{3}^{*M})$, consumer surplus is given by $CS _M^{\alpha _3<\alpha _{3}^{*M}}$ in expression (26) in the proof of Proposition 1, while when $\gamma \ge \frac{1}{3}$ and $\alpha _3 \in [{\alpha _{3}^{*M},\alpha _{3}(\gamma )}]$ the monopolist sells the cocktail only and consumer surplus is given by the expression (27)

Comparing consumer surplus with and without discrimination when $\gamma <\frac{1}{3}$ and when $\gamma \ge \frac{1}{3}$ but $\alpha _{3} \in (1, \alpha _{3}^{*M})$ so that in both cases drugs are sold as stand-alone treatments and in a cocktail, we see that

$$\begin{aligned} CS _M^{\alpha _3<\alpha _{3}^{*M}}- CS _M^d=\frac{(\alpha _3-1)^2}{12(1-\gamma )}>0. \end{aligned}$$

Thus, in this case discrimination entails a reduction in consumer surplus when $\gamma <\frac{1}{3}$ and $1<\alpha _3<\alpha _{3}^{*M}$.

Comparing consumer surplus with and without discrimination when $\gamma \ge \frac{1}{3}$ and $\alpha _3 \in [{\alpha _{3}^{*M},\alpha _{3}(\gamma )}]$ entails comparing a situation in which the consumer sells the cocktail only (with uniform pricing) and a situation in which the consumer sells the stand-alone treatments and the cocktail (with discrimination). We thus need to compare expression (27) to expression (33).

We see that

$$\begin{aligned} CS _M^d> CS _M^{\alpha _3>\alpha _{3}^{*M}} \end{aligned}$$

since

$$\begin{aligned} CS _M^d- CS _M^{\alpha _3>\alpha _{3i}^{*M}}=\frac{((\alpha _3+2) \gamma -3)^2}{36 (1-\gamma ) (3-2 \gamma )}>0 \end{aligned}$$

$\square$

Thus, discrimination entails a reduction in consumer surplus, for all $\gamma \in [0,1]$.

Proof of Proposition 5

Define $p_i^d$ and $p_{ic}^d$ ($i=1,2$) the price of drug i when it is sold as a stand-alone treatment and when it is a cocktail component, respectively. Conditional on both firms deciding to price discriminate, profits in the second stage would be $\varPi _{i}=p_{i}^d\,q_{i}+\frac{1}{2}\,p_{ic}^d\,q_{c}$, $i=1,2$.

Bertrand equilibrium prices are $p_{i}^{d}=\frac{3(1-\gamma )}{(6-5\gamma )}$, $p_{ic}^{d}=\frac{2}{3}\left( \alpha _3-\frac{\gamma }{6-5\gamma }\right)$. Quantities are $q_i^d=\frac{5 (\alpha _3+2) \gamma ^2-6 (\alpha _3+6) \gamma +27}{27 (\gamma -1) (5 \gamma -6)}$, $i=1,2$, $q_{c}^d=\frac{\alpha _3 (\gamma -3)+2 \gamma }{27 (\gamma -1)}$.

Demanded quantities of the stand-alone treatments are positive ($q_i^d>0$) if

$$\begin{aligned} \alpha _3<\alpha _{3i}^{*Dd}=\frac{27-36\gamma +10\gamma ^2}{\gamma (6-5\gamma )} \end{aligned}$$

(34)

However, it is easy to check that $\alpha _{3i}^{*Dd}>\alpha _3^*(\gamma )$, so that such quantities are always positive in our parameters’ range. Moreover, the demanded quantity of the cocktail is always positive under this pricing scheme, as well. In fact, we have $q_3^d>0$ if $\alpha _3>\frac{2\gamma }{3-\gamma }\le 1$ for all $\gamma \in [0,1]$.

Comparing prices, it can be seen that the cocktail will be sold at a premium $\delta ^{d}$, where $\delta ^{d}=p_{ic}^{d}-\{r_1 p_{1}^{d}+r_2 p_{1}\} =\frac{2}{3}\,\alpha _3-\frac{9-7\gamma }{3(6-5\gamma )}>0$.

Equilibrium profits are $\varPi _{i}^d=\frac{1}{81} \left( \frac{9 \left( 5 (\alpha _3+2) \gamma ^2-6 (\alpha _3+6) \gamma +27\right) }{(6-5 \gamma )^2}+\frac{(\alpha _3 (\gamma -3)+2 \gamma ) \left( \alpha _3+\frac{\gamma }{5 \gamma -6}\right) }{\gamma -1}\right)$, $i=1,2$.

We now compare $\varPi _{i}^{c}$, the profits without price discrimination we found in the proof of Proposition 3, with $\varPi _{i}^d$.

Particularly, we notice that $\varPi _{i}^{d}- \varPi _{i}^{c}>0$ if $\alpha _3>{\bar{\alpha }}_3(\gamma )=\frac{-738 \gamma ^3+2527 \gamma ^2-2892 \gamma +1107}{-360 \gamma ^3+1342 \gamma ^2-1662 \gamma +684}<\alpha _3(\gamma )$ so that both firms would engage in price discrimination if $\alpha _3$ is sufficiently high.

Also, ${\bar{\alpha }}_3(\gamma )\le 1$ if and only if $\gamma \ge 0.92$, which implies that, when the two stand-alone products are highly substitutable, price discrimination is profitable for any $\alpha _3 \ge 1$.

However, when $\gamma <0.92$, $\frac{d{\bar{\alpha }}_3}{d\gamma } < 0$, meaning that the less substitutable products are, the more likely it is that uniform pricing yields higher profits than price discrimination.

We now show that the ability to price discriminate might lead to a Prisoner’s dilemma, that is, to a situation in which both firms price discriminate even if they would be better off if they both did not and this because price discrimination is a dominant strategy.

If firm 1 decides to engage in price discrimination while firm 2 chooses uniform pricing, profits would be

$$\begin{aligned} \varPi _{1}& {}= p_{1}\,q_{1}+\frac{1}{2}\,p_{1c}\,q_{c}\nonumber \\ \varPi _{2}& {}= p_{2}\,\left( q_{2}+\frac{1}{2}\,q_{c}\right). \end{aligned}$$

(35)

Bertrand equilibrium prices are $p_{1}^{d^\prime }=\frac{(1-\gamma ) ((2 \alpha _3-23) \gamma +57)}{6 \left( 6 \gamma ^2-23 \gamma +19\right) }$, $p_{2}^{d^{\prime }}=\frac{2 (1-\gamma ) (\alpha _3 (3-2 \gamma )+4 (3-\gamma ))}{18 \gamma ^2-69 \gamma +57}$ and $p_{1c}^{d^\prime }=\frac{2 \alpha _3 \left( 7 \gamma ^2-31 \gamma +27\right) +\gamma ^2+5 \gamma -12}{18 \gamma ^2-69 \gamma +57}$, where the latter is the price firm 1 sets for a unit of its product purchased to be used in the cocktail.

Equilibrium quantities are $q_{1}^{d^{\prime }}=\frac{-4 (\alpha _3+2) \gamma ^3+6 (3 \alpha _3+8) \gamma ^2-(16 \alpha _3+95) \gamma +57}{18 (1-\gamma ) \left( 6 \gamma ^2-23 \gamma +19\right) }$, $q_{2}^{d^{\prime }}=\frac{(2-\gamma ) \left( \alpha _3 \left( 4 \gamma ^2-6\right) +8 \gamma ^2-39 \gamma +33\right) }{18 (1-\gamma ) \left( 6 \gamma ^2-23 \gamma +19\right) }$ for the stand-alone treatments and $q_{3}^{d^{\prime }}=\frac{\alpha _3 \left( -2 \gamma ^3+17 \gamma ^2-40 \gamma +27\right) -4 \gamma ^3+13 \gamma ^2-5 \gamma -6}{9 (1-\gamma ) \left( 6 \gamma ^2-23 \gamma +19\right) }$ for the cocktail.

It can be checked that $p_{i}^{d^{\prime }}>p_{i}^{d}$, $(i=1,2)$ and $p_{1c}^{d^{\prime }}>p_{ic}^{d}$, indicating that firm 1 would charge higher prices on both its stand-alone treatment and its cocktail component compared to the case in which both competitors price discriminate.

Substituting equilibrium prices and quantities into profits in (35), we can see that³³

$$\begin{aligned} \varPi _{1}^{d^{\prime }}>max\left\{ \varPi _{i}^{c},\varPi _{i}^{d}\right\} >\varPi _{2}^{d^{\prime }}. \end{aligned}$$

(36)

When $\gamma <0.92$ and $\alpha _3<{\bar{\alpha }}_3(\gamma )$, we know that $\varPi _{i}^{c}>\varPi _{i}^{d}$, so that the following holds:

$$\begin{aligned} \varPi _{1}^{d^{\prime }}>\varPi _{i}^{c}>\varPi _{i}^{d}>\varPi _{2}^{d^{\prime }}. \end{aligned}$$

(37)

Choosing to price discriminate in the first stage is a dominant strategy for both firms, given the symmetry of the game. The resulting equilibrium outcome is not Pareto-Efficient, though, given that $\varPi _{1}^{c}>\varPi _{i}^{d},i=1,2$. $\square$

Proof of Proposition 6

Consider first the case $\alpha _3=1$. Without price discrimination, consumer surplus is given by expression (29) in the proof of Proposition 2.

With non-coordinated price discrimination, using the definition of consumer surplus in expression (23), we find:

$$\begin{aligned} CS ^d_{\alpha _3=1}=\frac{25 \gamma ^2-82 \gamma +66}{18 (6-5 \gamma )^2} \end{aligned}$$

(38)

A direct comparison of expressions (29) and (38) shows that $CS ^c_{\alpha _3=1}> CS ^d_{\alpha _3=1}$ for all $\gamma \in [0,1)$. Surplus with and without price discrimination is the same only if $\gamma =1$, that is, when the two treatments are perfect substitutes. That is, however, a limit case, in which prices are set equal to marginal costs (here zero).

We then turn to the case in which $\alpha _3>1$.

Using again expression (23), when $1<\alpha _3<\alpha _3^*(\gamma )$, consumer surplus under uniform pricing equals expression (30) in the proof of Proposition 3.

With price discrimination, consumer surplus equals

$$\begin{aligned} CS ^d_{\alpha _3>1}=\frac{\alpha _3^2 (6-5 \gamma )^2 (\gamma -3)+4 \alpha _3 (6-5 \gamma )^2 \gamma +100 \gamma ^3-588 \gamma ^2+972 \gamma -486}{162 (6-5 \gamma )^2 (\gamma -1)} \end{aligned}$$

(39)

Computing the difference between (30) and (39) we find

$$\begin{aligned}&CS _d^{\alpha _3}- CS ^c_{\alpha _3>1} \nonumber \\&\quad =\frac{\left( 1710 \alpha _3 \gamma ^3-6232 \alpha _3 \gamma ^2+7566 \alpha _3 \gamma -3060 \alpha _3-1683 \gamma ^3+6031 \gamma ^2-7221 \gamma +2889\right) (\alpha _3 (10 \gamma -12)-7 \gamma +9)}{81 (1-\gamma ) (5 \gamma -6)^2 (9 \gamma -11)^2} \end{aligned}$$

(40)

The denominator in expression (40) is always positive. We thus focus on the sign of the numerator. First of all, notice that (40) is increasing in $\alpha _3$ iff:

$$\begin{aligned} \alpha _3>\frac{8 \left( 18 \gamma ^2-44 \gamma +27\right) }{171 \gamma ^2-418 \gamma +255} \end{aligned}$$

(41)

Since the right-hand-side of (41) is always smaller than 1, this condition is always satisfied, given our assumptions ($\alpha _3>1$). Thus, the difference $CS ^c_{\alpha _3>1}- CS ^d_{\alpha _3>1}$ is always increasing in $\alpha _3$ and to prove that $CS ^c_{\alpha _3>1}- CS ^d_{\alpha _3>1}>0$ it suffices that $CS ^c_{\alpha _3>1}- CS ^d_{\alpha _3>1}>0$ for $\alpha _3=1$. In fact, we have shown before that, for $\alpha _3=1$, $CS ^c_{\alpha _3=1}> CS ^d_{\alpha _3=1}$ always. Thus, $CS ^c_{\alpha _3>1}> CS ^d_{\alpha _3>1}$ for all $\gamma \in [0,1)$ and for all $1<\alpha _3<\alpha _3^*(\gamma )$.

Proof of Lemma 3

When $1<\alpha _3 <\alpha _3^*(\gamma )$, we need to compare expression (39) with the amount of consumer surplus without the cocktail (28).

$CS ^d_{\alpha _3>1}$ is an increasing function of $\alpha _3$, whereas $CS ^{nc}$ is invariant with respect to $\alpha _3$.³⁴

At $\alpha _3=1$, $CS ^d_{\alpha _3>1}< CS ^{nc}$.

At $\alpha _3=\alpha _3^*(\gamma )$, $CS ^d_{\alpha _3>1}> CS ^{nc}$ if $\gamma <0.69$.

Thus, if $\gamma \ge 0.69$, $CS ^d_{\alpha _3>1}\le CS ^{nc}$ at $\alpha _3=\alpha _3^*(\gamma )$.

If $\gamma <0.69$, there exists ${\hat{\alpha }}_3$, such that $CS ^d_{\alpha _3>1}> CS ^{nc}$ if and only if $\alpha _3>{\hat{\alpha }}_3$ and $CS ^d_{\alpha _3>1}< CS ^{nc}$ if and only if $\alpha _3<{\hat{\alpha }}_3$. $\square$

Proof of Proposition 7

We solve the game by backward induction. Given $\delta$ and k, each firm maximizes $\varPi _i$ ($i=1,2$) after substituting demand expressions from (9) to (11). We thus obtain the Bertrand equilibrium prices:

$$\begin{aligned} p_{1}^{\delta }=\frac{2(1-\gamma )(9(3-\gamma )+\delta (k(11-9\gamma )+8))}{3\left( 33-38\gamma +9\gamma ^{2}\right) } \end{aligned}$$

(42)

$$\begin{aligned} p_{2}^{\delta }=\frac{2(1-\gamma )(9(3-\gamma )+\delta (19-9\gamma -k(11-9\gamma )))}{3\left( 33-38\gamma +9\gamma ^{2}\right) }. \end{aligned}$$

(43)

Using these equilibrium prices, and assuming $k=\frac{1}{2}$ we obtain the equilibrium quantities:

$$\begin{aligned} q_i^{\delta }=\frac{15+9 \gamma ^2 -24 \gamma +\delta (7 \gamma -9)}{81 \gamma ^2-180 \gamma +99},\;\;i=1,2 \end{aligned}$$

(44)

$$\begin{aligned} q_{3}^{\delta }=\frac{15+9 \gamma ^2 -24 \gamma +4 \delta (6 -5\gamma )}{81 \gamma ^2-180 \gamma +99}. \end{aligned}$$

(45)

Finally, substituting these equilibrium values in the profit functions (20) and (21), we obtain the equilibrium profits as a function of $\delta$:

$$\begin{aligned}&\varPi _{i}^{\delta }=\frac{9 \gamma ^2 \left( -9 \delta ^2+\delta +33\right) -\gamma \left( -200 \delta ^2+12 \delta +351\right) +3 \left( -41 \delta ^2+\delta +45\right) -81 \gamma ^3}{9 (11-9 \gamma )^2 (1-\gamma )},\nonumber \\&\quad i=1,2. \end{aligned}$$

(46)

Maximizing joint profits $\varPi _{1}^{\delta }+\varPi _{2}^{\delta }$ with respect to $\delta$, we obtain

$$\begin{aligned} \delta ^{*}=\frac{3 (1-3 \gamma ) (1-\gamma )}{162 \gamma ^2-400 \gamma +246} \end{aligned}$$

(47)

from which it is immediate to see that $\delta ^{*}>0$ when $\gamma <\frac{1}{3}$ and $\delta ^{*}<0$ when $\gamma >\frac{1}{3}$. $\square$

Proof of Lemma 4

Substituting $\delta ^{*}$ in the Bertrand–Nash equilibrium prices, quantities and profits we obtain

$$\begin{aligned} p_{i}^{\delta *}=\frac{3 (45-37 \gamma ) (1-\gamma )}{162 \gamma ^2-400 \gamma +246},\;\;i=1,2 \end{aligned}$$

(48)

$$\begin{aligned} p_{3}^{\delta *}=\frac{51 \gamma ^2-117 \gamma +66}{81 \gamma ^2-200 \gamma +123} \end{aligned}$$

(49)

$$\begin{aligned} q_{i}^{\delta *}=\frac{54 \gamma ^2-155 \gamma +111}{486 \gamma ^2-1200 \gamma +738},\;\;i=1,2 \end{aligned}$$

(50)

$$\begin{aligned} q_{3}^{\delta *}=\frac{27 \gamma ^2-82 \gamma +57}{243 \gamma ^2-600 \gamma +369} \end{aligned}$$

(51)

and

$$\begin{aligned} \varPi _{i}^{\delta *}=\frac{(61-36 \gamma ) (1-\gamma )}{324 \gamma ^2-800 \gamma +492},\;\;i=1,2 \end{aligned}$$

(52)

• a. Comparing the price in (48) to the price set absent price discrimination (obtained in Proposition 2), $p_i^c=\frac{6(1-\gamma )}{11-9\gamma }$, we can see that $p_{i}^{\delta *}>p_i^c$ if and only if $\gamma <\frac{1}{3}$.

  Comparing the price of the cocktail with coordinated price discrimination [in expression (49)], with the cocktail price without discrimination (obtained in Proposition 2 and equal to $p_3^c=r_1 p_1^c+r_2 p_2^c$), we can see that $p_{3}^{\delta *}<p_3^c$ if and only if $\gamma <\frac{1}{3}$.

• b. Similarly, comparing (48) and (49) to the prices set with non-cooperative price discrimination (obtained in Proposition 5), $p_i^d=\frac{3(1-\gamma )}{6-5\gamma }$ ($i=1,2$), and $p_{3c}^{d}=\frac{2}{3}\left( \alpha _3-\frac{\gamma }{6-5\gamma }\right)$ it is possible to prove that $p_{i}^{\delta *}>p_i^d$ ($i=1,2$) and $p_{3}^{\delta *}<p_3^d$ for all $\gamma \in [0,1)$.

• c. Comparing profits, it is possible to check that $\varPi _{i}^{\delta *}>\varPi _{i}^{cd}>\varPi _{i}^{c}$, that is, the profits under coordinated price discrimination are always higher than those under non-coordinated price discrimination, which, in turn, are higher than profits without discrimination, and this for all $\gamma \in [0,1)$$\square$

Proof of Proposition 8

Substituting equilibrium prices and quantities into the general expression for consumer surplus (23), consumer surplus with coordinated price discrimination is

$$\begin{aligned} CS ^{\delta *}=\frac{18 \gamma ^2-61 \gamma +51}{324 \gamma ^2-800 \gamma +492}. \end{aligned}$$

(53)

• a. Comparing Eq. (53) with the expression for consumer surplus under uniform pricing, $CS ^c$ (obtained in Proposition 2) we find that $CS ^{\delta *}\ge CS ^c$ if $\gamma \le \frac{1}{3}$.

• b. Similarly, comparing Eq. (53) with the expression for consumer surplus under non-coordinated price discrimination $CS ^d_{\alpha _3=1}$ found in expression (38) in the proof of Proposition 6, we find that $CS ^{\delta *} > CS ^d_{\alpha _3=1}$ for all $\gamma \in [0,1)$.

$\square$

Proof of Proposition 9

Following the same steps described in the proof of Proposition 7, the value of $\delta$ that maximizes joint profits is $\delta ^{*}=\frac{12-206\gamma +87\gamma ^{2}-2a_{3}(60-97\gamma +39\gamma ^{2})}{(246-400\gamma +162\gamma ^{2})}$. Notice that $\frac{ \partial \delta ^{*}}{\partial \alpha _{3}}<0$ for any $\gamma \in [0,1)$, and that $\delta ^{*}\lesseqgtr 0$ iff $\gamma \gtreqless {\widetilde{\gamma }}(\alpha _{3})=\frac{97a_{3}-103+\sqrt{ 49a_{3}^{2}+52a_{3}-92}}{3(26a_{3}-29)}$, where $\frac{d{\widetilde{\gamma }} (\alpha _{3})}{d\alpha _{3}}<0$. Finally, ${\widetilde{\gamma }}(\alpha _{3})=0$ when $\alpha _{3}=1.025$, so that for any $\alpha _{3}\ge 1.025$, firms would only coordinate on a premium no matter the degree of substitutability across single products.

Notice that, in this case,

$$\begin{aligned} \alpha _{3i}^{*Dcoord}=\frac{369 -108 \gamma ^3+517 \gamma ^2-774 \gamma }{54 \gamma ^3-110 \gamma ^2+24 \gamma +36}. \end{aligned}$$

(54)

$\square$