Endogenous timing in a mixed duopoly

Abstract

This paper applies the framework of endogenous timing in games to mixed quantity duopoly, wherein a private—domestic or foreign—firm competes with a public, welfare-maximizing firm. A central goal of the paper is to present a unified and general treatment of the basic question of what constitutes the appropriate solution concept—Cournot or Stackelberg—in such duopolies. We show that simultaneous play never emerges as a subgame-perfect equilibrium of the extended game, in sharp contrast to private duopoly games. We demonstrate that this result is due to the objective function of the public firm being increasing in the rival’s output (instead of decreasing for a private firm). We provide sufficient conditions for the emergence of public and/or private leadership equilibrium. In all cases, private profits and social welfare are higher than under the corresponding Cournot equilibrium. We make extensive use of the basic results from the theory of supermodular games in order to avoid common extraneous assumptions such as concavity, existence and uniqueness of the different equilibria, whenever possible. Some policy implications are drawn, in particular those relating to the merits of privatization.

Appendix

The Appendix includes most of the proofs of the paper but also some intermediate results that are needed for the proof of the main theorems of the paper.

Proof of Lemma 1

A sufficient condition for $W^{f}\left( q_{0},q_{1}\right)$ to have the strict IDP (strict DDP) is $\frac{\partial ^{2}W^{f}\left( q_{0},q_{1}\right) }{\partial q_{0}\partial q_{1}}>(<)0$. By the strengthening of Topkis’s Theorem given in Amir (1996b) or Edlin and Shannon (1998), a sufficient condition for every selection of $r_{0}\left( .\right)$ to be strictly increasing (strictly decreasing) whenever interior is for $\frac{\partial W^{f}\left( q_{0},q_{1}\right) }{ \partial q_{0}}$ to be strictly increasing (decreasing) in $q_{1}$. The latter condition is in turn implied by $P^{\prime \prime }\left( q_{0}+q_{1}\right) <(>)0$ since

$$\begin{aligned} \frac{\partial ^{2}W^{f}\left( q_{0},q_{1}\right) }{\partial q_{0}\partial q_{1}}=-q_{1}P^{\prime \prime }\left( q_{0}+q_{1}\right) . \end{aligned}$$

$\square$

Proof of Lemma 2

• (a) A sufficient condition for the private firm’s objective to have the strict DDP is (see Novshek 1985)

  $$\begin{aligned} \frac{\partial ^{2}\varPi _{1}\left( q_{0},q_{1}\right) }{\partial q_{0}\partial q_{1}}=P^{\prime }\left( q_{0}+q_{1}\right) +q_{1}P^{\prime \prime }\left( q_{0}+q_{1}\right) <0,\quad \forall q_{0},q_{1}\ge 0 \end{aligned}$$

  which is equivalent to

  $$\begin{aligned} P^{\prime }\left( x\right) +xP^{\prime \prime }\left( x\right) <0,\quad \forall x\ge 0 \end{aligned}$$

  A sufficient condition for the strict dual SSCP is

  $$\begin{aligned} P\left( x\right) P^{\prime \prime }\left( x\right) -P^{\prime 2}\left( x\right) <0,\quad \forall x\ge 0 \end{aligned}$$

  as defined in Amir (1996a). If one of these conditions holds, then any selection of the private firm’s best-response correspondence is strictly decreasing (Amir 1996b).

• (b) Since the production costs of the private firm are zero, its objective is $q_{1}P\left( q_{0}+q_{1}\right)$. Given that $\log P\left( \cdot \right)$ is convex by assumption, it follows that the objective is strictly log-supermodular, i.e.

  $$\begin{aligned} \frac{\partial ^{2}\log q_{1}P\left( q_{0}+q_{1}\right) }{\partial q_{0}\partial q_{1}}=\frac{P\left( q_{0}+q_{1}\right) P^{\prime \prime }\left( q_{0}+q_{1}\right) -P^{\prime 2}\left( q_{0}+q_{1}\right) }{ P^{2}\left( q_{0}+q_{1}\right) }>0. \end{aligned}$$

So, every selection of the private firm best-response correspondence is strictly increasing (Amir 1996b). $\square$

Before starting the proof of Theorem 1 we need to establish some intermediate results in the following Lemmas 4, 5, and 6.

Lemma 4

Consider the case of a foreign private firm. Assume the game if of strict strategic substitutes. Then the set N includes a point $\left( \underline{q_{0}},\overline{q_{1}}\right)$, with $\underline{q_{0}}$ being the public firm’s smallest output and  $\overline{ q_{1}}$ the private firm’s largest output in the set N, which is the Pareto dominant mixed-duopoly Cournot equilibrium.

Proof

Reversing the natural order on the public firm’s action set, the set $N$ has a largest Cournot equilibrium in the new order, which is clearly $\left( { \underline{q_{0}},\overline{q_{1}}}\right)$.²¹ We show that this is the most preferred Cournot equilibrium by both private and public firms. Consider any $\left( \widehat{q}_{0},\widehat{q}_{1}\right) \in N$. By the extremal nature of $\left( \underline{q_{0}},\overline{q_{1}}\right)$ it follows that $\widehat{q}_{1}\le \overline{q_{1}}$ and $\widehat{q}_{0}\ge \underline{q_{0}}$.

• (i) For the private firm the following relation holds:

  $$\begin{aligned} \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge \varPi _{1}\left( {\underline{q_{0}},\widehat{q}_{1}}\right) \ge \varPi _{1}\left( \widehat{q} _{0},\widehat{q}_{1}\right) \end{aligned}$$

  (7)

  where the first inequality derives from $\overline{q_{1}}$ being best response to $\underline{q_{0}}$, and the second inequality is due to private firm’s objective being strictly decreasing in $q_{0}$.

• (ii) For the public firm, we have

  $$\begin{aligned} W^{f}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge W^{f}\left( \widehat{q}_{0},\overline{q_{1}}\right) \ge W^{f}\left( \widehat{q}_{0}, \widehat{q}_{1}\right) \end{aligned}$$

  (8)

  where the first inequality is due to the fact that $\underline{q_{0}}$ is a best response to $\overline{q_{1}}$ and the second derives from $W^{f}$ being strictly increasing in $q_{1}$.

This completes the proof. $\square$

Lemma 5

Consider the case of a foreign private firm. Assume the game is of strict strategic substitutes. Then the output of the private firm (public firm) in a Stackelberg game under either order of moves is larger (smaller) than in the Pareto-dominant mixed-duopoly Cournot equilibrium.

Proof

We have to distinguish the two cases of (i) private leadership, and (ii) public leadership.

• (i) Suppose, towards contradiction, that there exists a point $\left( q_{0}^{f},q_{1}^{l}\right) \in S_{1}$ such that $q_{1}^{l}<\overline{q_{1}}$. Since both $\left( q_{0}^{f},q_{1}^{l}\right)$ and $\left( \underline{ q_{0}},\overline{q_{1}}\right)$ lay on $GR\,r_{0}(.)$, which has only strictly decreasing selections, it follows that $q_{0}^{f}>\underline{q_{0}}$. Then

  $$\begin{aligned} \varPi _{1}\left( {\underline{q_{0}},\overline{q_{1}}}\right) \ge \varPi _{1}\left( {\underline{q_{0}},q_{1}^{l}}\right) >\varPi _{1}\left( q_{0}^{f},q_{1}^{l}\right) \end{aligned}$$

  (9)

  where the first inequality is due to $\overline{q_{1}}$ being a best response to $\underline{q_{0}}$ and the second by the fact that $\varPi _{1}$ is strictly decreasing in $q_{0}$. Inequality (9) contradicts the nature of Stackelberg equilibrium. Hence it is always the case that $q_{1}^{l}\ge \overline{q_{1}}$ and $q_{0}^{f}\le \underline{ q_{0}}$.

• (ii) Suppose, towards contradiction, that there exists a point $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ such that $q_{0}^{l}>\underline{ q_{0}}$, and so $q_{1}^{f}<\overline{q_{1}}$ since $r_{1}$ is strictly decreasing. Then, the following inequalities hold:

  $$\begin{aligned} W^{f}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge W^{f}\left( {q} _{0}^{l},\overline{q_{1}}\right) >W^{f}\left( q_{0}^{l},q_{1}^{f}\right) \end{aligned}$$

  (10)

  where the first inequality is due to $\underline{q_{0}}$ being a best response to $\overline{q_{1}}$ and the second to the fact that $W^{f}\left( \cdot \right)$ is strictly increasing in $q_{1}$. Inequality (10) contradicts the nature of Stackelberg equilibria and so it is always the case that $q_{0}^{l}\le \underline{q_{0}}$ and $q_{1}^{f}\ge \overline{q_{1}}$.

This completes the proof. $\square$

Lemma 6

Assume the game is of strict strategic substitutes. Then the payoff of the leader in any Stackelberg equilibrium is strictly larger than in any Cournot equilibrium as long as the Pareto dominant Cournot equilibrium is interior.

Proof

If $\left( \underline{q_{0}},\overline{q_{1}}\right)$ is interior, the following first-order conditions for private and public firms must hold:

$$\begin{aligned} \frac{\partial \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) }{ \partial q_{1}}=0; \qquad \frac{\partial W^{i}\left( \underline{q_{0}}, \overline{q_{1}}\right) }{\partial q_{0}}=0 \quad i=d,f \end{aligned}$$

We analyze the two possible Stackelberg games separately again.

• (i) By Remark 1 in Sect. 3.2, with private leadership, the private firm’s objective can be shown to be $\varPi _{1}\left( \underline{r_{0}}(q_{1}),q_{1}\right)$ where $\underline{ r_{0}}(\cdot )$ is the minimal selection from the public firm’s optimal reaction correspondence (see Amir and Grilo 1999 for a proof). By the Maximum Theorem, the correspondence $r_{0}$ has a closed graph. Hence, $\underline{r_{0}}$ is lower semi-continuous and right-continuous (for a proof, see Amir 1996b). Since $\varPi _{1}$ is decreasing in its first argument and $\underline{r_{0}}$ is decreasing and right-continuous, $\varPi _{1}\left( \underline{r_{0}}\left( q_{1}^{l}\right) ,q_{1}^{l}\right)$ can have only upward jumps in $q_{1}$ (and no downward jumps) and is right-continuous in $q_{1}$. Hence, if $\left( q_{0}^{f},q_{1}^{l}\right) \in S_{1}$ is also interior, then the following first-order condition for the private leader must hold

  $$\begin{aligned} \frac{\partial \varPi _{1}\left( q_{0}^{f},q_{1}^{l}\right) }{\partial q_{1}}+ \frac{\partial \varPi _{1}\left( q_{0}^{f},q_{1}^{l}\right) }{\partial q_{0}}{ \underline{r_{0}}^{\prime }}\left( q_{1}^{l}\right) \le 0 \end{aligned}$$

  (11)

  where $r_{0}^{\prime }\left( q_{1}^{l}\right)$ is a right Dini derivate.²² Since $\frac{\partial \varPi _{1}\left( { \underline{q_{0}}\overline{q_{1}}}\right) }{\partial q_{0}}<0$ by Eq. (5), and $\underline{r_{0}}^{\prime }\left( q_{1}^{l}\right) =-\frac{\partial ^{2}W^{i}\left( {q}_{0},\overline{q_{1}}\right) /\partial q_{0}\partial q_{1}}{\partial ^{2}W^{i}\left( {q}_{0},\overline{q_{1}} \right) / \partial q_{0}^{2}}<0$ whenever $\underline{r_{0}}$ is interior by point (a) of Lemma 1, it follows from (11) that $\frac{\partial \varPi _{1}\left( q_{0}^{f},q_{1}^{l}\right) }{\partial q_{1}}<0$ at any interior point. Hence, $\left( \underline{q_{0}},\overline{ q_{1}}\right) \not \in S_{1}$, andhence $q_{1}^{l} > \overline{q_{1}}$ and $q_{0}^{f} < \underline{ q_{0}}$. If $\left( q_{0}^{f},q_{1}^{l}\right)$ is not interior, then the conclusion follows from the interiority of $\left( \underline{q_{0}},\overline{q_{1}} \right)$.

• (ii) With public leadership, the public firm’s objective can be shown to be $W^{i}\left( q_{0},\overline{r_{1}}(q_{0})\right)$ where $\overline{ r_{1}}(\cdot )$ is the maximal selection from the public firm’s optimal reaction correspondence (see Amir and Grilo 1999 for a proof). Since the point $\left( q_{0}^{l},q_{1}^{f}\right)$ lies on $\overline{r_{1}}\left( \cdot \right)$, the following first-order condition of the private firm holds:

  $$\begin{aligned} P\left( q_{0}+\overline{r_{1}}\left( q_{0}\right) \right) +\overline{r_{1}} \left( q_{0}\right) P^{\prime }\left( q_{0}+\overline{r_{1}}\left( q_{0}\right) \right) -C_{1}^{\prime }\left( \overline{r_{1}}\left( q_{0}\right) \right) =0 \end{aligned}$$

  Then, at $\left( q_{0}^{l},q_{1}^{f}\right)$, price is strictly above the private firm’s marginal cost and it follows that $\frac{\partial W^{i}\left( q_{0}^{l},q_{1}^{f}\right) }{\partial q_{1}}>0$. Using also the fact that $\overline{r_{1}}$ is decreasing and left-continuous (as in Part (i)), we know that $W^{i}\left( q_{0},\overline{r_{1}}(q_{0})\right)$ can have only downward jumps in $q_{0}$ (and no upward jumps) and is left-continuous in $q_{0}$. Hence, if $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ is interior, the following first-order condition for the public leader must hold (here, $r_{1}^{\prime }\left( q_{0}^{l}\right)$ is a left Dini derivate)

  $$\begin{aligned} \frac{\partial W^{i}\left( q_{0}^{l},q_{1}^{f}\right) }{\partial q_{0}}+ \frac{\partial W^{i}\left( q_{0}^{l},q_{1}^{f}\right) }{\partial q_{1}} r_{1}^{\prime }\left( q_{0}^{l}\right) \ge 0,\quad i=f,d. \end{aligned}$$

  (12)

  Since $\left( q_{0}^{l},q_{1}^{f}\right)$ is interior, we have $r_{1}^{\prime }(q_{0}^{l})<0$ via an argument similar to the analogous step in part (i). Then (12) implies that $\frac{\partial W^{i}\left( q_{0}^{l},q_{1}^{f}\right) }{\partial q_{0}}>0$. This implies that $\left( { \underline{q_{0}},\overline{q_{1}}}\right) \not \in S_{0}$. Therefore $q_{0}^{l}<\underline{q_{0}}$ and $q_{1}^{f}>\overline{q_{1}}$ . If $\left( q_{0}^{l},q_{1}^{f}\right)$ is not interior, then the conclusion follows from the interiority of $\left( \underline{q_{0}},\overline{q_{1}} \right)$.

This completes the proof. $\square$

Proof of Theorem 1

In Lemma 4 we have shown that the point $\left( \underline{q_{0}},\overline{q_{1}}\right)$ is the most preferred simultaneous-move equilibrium by both firms. Moreover, by Lemmas 5–6, any $\left( q_{0}^{f},q_{1}^{l}\right) \in S_{1}$ is such that $q_{1}^{l}>\overline{q_{1} }$ and $q_{0}^{f}<\underline{q_{0}}$, and any $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ is such that $q_{1}^{f}>\overline{q_{1} }$ and $q_{0}^{l}<\underline{q_{0}}$. Now we show that each firm strictly prefers being a follower in a sequential-move game to the Pareto dominant Cournot equilibrium.

For the private firm, the following relation holds

$$\begin{aligned} \varPi _{1}\left( q_{0}^{l},q_{1}^{f}\right) \ge \varPi _{1}\left( q_{0}^{l}, \overline{q_{1}}\right) >\varPi _{1}\left( \underline{q_{0}},\overline{q_{1}} \right) \end{aligned}$$

where the first inequality comes from the fact that $q_{1}^{f}$ is a best response to $q_{0}^{l}$ and the second is due to the private firm’s objective being strictly decreasing in $q_{0}$.

For the public firm the following relation holds:

$$\begin{aligned} W^{f}\left( q_{0}^{f},q_{1}^{l}\right) \ge W^{f}\left( \underline{q_{0}}, q_{1}^{l}\right) >W^{f}\left( \underline{q_{0}},\overline{q_{1}}\right) \end{aligned}$$

where the first inequality is due to $q_{0}^{f}$ being a best response to $q_{1}^{l}$, and the second derives from the fact that W is strictly increasing in $q_{1}$.

Then, the conditions of point (ii) of Proposition 1 hold and both Stackelberg equilibria are SPEs of the endogenous timing game. $\square$

Proof of Theorem 2

Since the reaction correspondences of the two firms are strictly monotonic in opposite directions when interior, the postulated mixed duopoly Cournot equilibrium must be unique. Call it $\left( \underline{q_{0}}, \overline{q_{1}}\right)$. We now prove that: (i) the public firm is better off in the private leadership than in the mixed-duopoly Cournot equilibrium; while (ii) the private firm’s payoff is strictly larger in the Cournot than in the public leadership equilibrium.

• (i) For the private leadership equilibrium, since $r_{0}$ is strictly decreasing in the interior, we can apply the same analysis as in point $(i)$ of the proofs of Lemmas 5–6 to show that $q_{1}^{l}>\overline{q_{1}}$ and $q_{0}^{f}<\underline{q_{0}}$. It follows that

  $$\begin{aligned} W^{f}\left( q_{0}^{f},q_{1}^{l}\right) \ge W^{f}\left( \underline{q_{0}}, q_{1}^{l}\right) >W^{f}\left( \underline{q_{0}},\overline{q_{1}}\right) \end{aligned}$$

  where the first inequality is due to $q_{0}^{f}$ being best response to $q_{1}^{l}$, and the second derives from the fact that $W_{f}$ is strictly increasing in $q_{1}$.

• (ii) We now analyze the public firm’s leadership equilibrium. First we show that the Stackelberg equilibrium $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ is such that $q_{0}^{l}\!>\!\underline{q_{0}}$ and $q_{1}^{f}\!>\! \overline{q_{1}}$. Suppose by contradiction that $q_{0}^{l}<\underline{q_{0}}$. Since $r_{1}$ is strictly increasing when interior, it follows that $q_{1}^{f}<\overline{q_{1}}$. Then,

  $$\begin{aligned} W^{f}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge W^{f}\left( q_{0}^{l},\overline{q_{1}}\right) >W^{f}\left( q_{0}^{l},q_{1}^{f}\right) \end{aligned}$$

  where the first inequality comes from $\underline{q_{0}}$ being a best response to $\overline{q_{1}}$ and the second is due to $W^{f}\left( q_{0},q_{1}\right)$ being strictly increasing in $q_{1}$. Therefore $q_{0}^{l}\ge \underline{q_{0}}$. Moreover, by an argument analogous to the one used in point $(i)$ of the proof of Lemma 6, $\left( \underline{q_{0}},\overline{q_{1}}\right) \not \in S_{0}$ and therefore $q_{0}^{l}>\underline{q_{0}}$. Since $r_{1}$ is strictly increasing in the interior, it follows that $q_{1}^{f}>\overline{q_{1}}$. As a consequence, we can rank the private firm’s payoffs in the following way:

  $$\begin{aligned} \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge \varPi _{1}( \underline{q_{0}},q_{1}^{f})>\varPi _{1}(q_{0}^{l},q_{1}^{f}) \end{aligned}$$

  where the first inequality follows from the property of Cournot equilibria and the second from the fact that $\varPi _{1}$ is strictly decreasing in $q_{0}$. Hence, the private firm strictly prefers Cournot to public leadership equilibria.

The conditions of point (iii) of Proposition 1 hold and the private leadership equilibrium is the unique SPE of the endogenous timing game. $\square$

Proof of Theorem 3

As in the previous Theorem, the postulated mixed duopoly Cournot equilibrium must be unique (since $r_{0}\left( q_{1}\right)$ is strictly increasing and $r_{1}\left( q_{0}\right)$ is strictly decreasing in the interior). Call it $\left( \underline{q_{0}},\overline{q_{1}}\right)$. We now prove that: (i) the public firm’s payoff is strictly larger in the Cournot equilibrium that in the private leadership equilibrium; while (ii) the private firm is better off in the public leadership equilibrium than in the Cournot equilibrium.

• (i) First we show that the private leadership equilibrium is such that $q_{1}^{l}<\overline{q_{1}}$ and $q_{0}^{f}<\underline{q_{0}}$. Suppose by contradiction that $q_{1}^{l}>\overline{q_{1}}$; since $r_{0}$ is strictly increasing in the interior $q_{0}^{f}>\underline{q_{0}}$. It follows that

  $$\begin{aligned} \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge \varPi _{1}\left( \underline{q_{0}},q_{1}^{l}\right) >\varPi _{1}\left( q_{0}^{f},q_{1}^{l}\right) \end{aligned}$$

  where the first inequality is due to $\overline{q_{1}}$ being a best response to $\underline{q_{0}}$, and the second derives from the fact that $\varPi _{1}\left( q_{0},q_{1}\right)$ is strictly decreasing in $q_{0}$. Therefore $q_{1}^{l}\le \overline{q_{1}}$ and $q_{0}^{f}\le \underline{q_{0} }$. By an argument analogous to the one in point $(i)$ of the proof of Lemma 6, $\left( \underline{q_{0}},\overline{q_{1}}\right) \not \in S_{1}$ and therefore $q_{1}^{l}<\overline{q_{1}}$ and $q_{0}^{f}< \underline{q_{0}}$. Furthermore, we have

  $$\begin{aligned} W^{f}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge W^{f}\left( q_{0}^{f},\overline{q_{1}}\right) >W^{f}\left( q_{0}^{f},q_{1}^{l}\right) \end{aligned}$$

  where the first inequality is due to $\underline{q_{0}}$ being a best response to $\overline{q_{1}}$ and the second to $W^{f}\left( q_{0},q_{1}\right)$ being strictly increasing in $q_{1}$.

• (ii) From the interiority of $\left( \underline{q_{0}},\overline{q_{1} }\right)$ and since $r_{1}$ is strictly decreasing in the interior, we can apply the same analysis as point $(ii)$ in the proof of Lemma 5 to show that $q_{0}^{l}\le \underline{q_{0}}$ and $q_{1}^{f}\ge \overline{q_{1}}$. Moreover, by point $(ii)$ in the proof of Lemma 6, $\left( \underline{q_{0}},\overline{q_{1}} \right) \not \in S_{0}$ and therefore $q_{0}^{l}<\underline{q_{0}}$ and $q_{1}^{f}>\overline{q_{1}}$. It follows that

  $$\begin{aligned} \varPi _{1}\left( q_{0}^{l},q_{1}^{f}\right) \ge \varPi _{1}\left( q_{0}^{l}, \overline{q_{1}}\right) >\varPi _{1}\left( \underline{q_{0}},\overline{q_{1}} \right) \end{aligned}$$

  where the first inequality follows from the fact that $q_{1}^{f}$ is a best response to $q_{0}^{l}$ and the second from the fact that $\varPi _{1}\left( q_{0},q_{1}\right)$ is strictly decreasing in $q_{0}$. Then, the private firm strictly prefers public leadership to the best Cournot equilibrium.

The conditions of point (iii) of Proposition 1 hold and public leadership is the unique SPE of the endogenous timing game. $\square$

Proof of Lemma 3

Since the public firm maximizes total welfare as defined in Eq. (1), its objective has DDP since

$$\begin{aligned} \frac{\partial ^{2}W^{d}\left( q_{0},q_{1}\right) }{\partial q_{0}\partial q_{1}}=P^{\prime }\left( q_{0}+q_{1}\right) <0 \end{aligned}$$

So, under the standard assumptions, any selection of the best-response correspondence of the public firm is strictly decreasing. $\square$

In order to prove Theorem 4, we need to establish some intermediate results in the following Lemmas 7 and 8.

Lemma 7

Consider the case of a domestic private firm. A sufficient condition for the extremal equilibrium $\left( \underline{q_{0}}, \overline{q_{1}}\right)$ to be the Pareto dominant mixed-duopoly Cournot equilibrium is that the public firm’s cost function is convex in the relevant range.

Proof

As in Lemma 4, since both reaction correspondences are strictly decreasing in the interior, $\left( \underline{ q_{0}},\overline{q_{1}}\right)$ is the extremal point in $N$ and any other $\left( \widehat{q}_{0},\widehat{q}_{1}\right) \in N$ distinct from $\left( \underline{q_{0}},\overline{q_{1}}\right)$ is such that $\widehat{q}_{1}< \overline{q_{1}}$ and $\widehat{q}_{0}>\underline{q_{0}}$.

• (i) For the private firm, we have that $\varPi _{1}\left( \underline{q_{0} },\overline{q_{1}}\right) >\varPi _{1}\left( \widehat{q}_{0},\widehat{q} _{1}\right)$, as shown in Lemma 4 by the inequality chain (7);

• (ii) For the public firm, we first show that the slopes of $r_{0}\left( q_{1}\right)$ are all in $[-1,0)$. To this end, consider the change of variable $x=q_{0}+q_{1}$ and view $x$ as the new decision variable for the public firm in the modified objective

  $$\begin{aligned} \max _{x\ge 0}\int \limits _{0}^{x}P\left( t\right) dt-C_{0}\left( x-q_{1}\right) -C_{1}\left( q_{1}\right) \end{aligned}$$

  Since the cross-partial of this objective w.r.t. $x$ and $q_{1}$ is equal to $C_{0}^{\prime \prime }\left( x-q_{1}\right) \ge 0$, by the convexity assumption, we conclude that the argmax $x^{*}(q_{1})$ is increasing in $q_{1}\ ($note here that $x^{*}(q_{1})$ is single-valued since $W^{d}$ is strictly concave in $q_{0}$). Since $x^{*}(q_{1})=r_{0}(q_{1})+q_{1}$, we conclude that the slopes of $r_{0}(q_{1})$ are all $\ge -1$. A standard application of the smooth Implicit Function Theorem shows that $r_{0}^{\prime }(q_{1})\ge -1$. Given that both $\left( \widehat{q}_{0},\widehat{q}_{1}\right)$ and $\left( \underline{q_{0}},\overline{q_{1}}\right)$ lie on $GR\,r_{0}\left( q_{1}\right)$, it follows that²³

  $$\begin{aligned} W^{d}\left( \underline{q_{0}},\overline{q_{1}}\right) -W^{d}\left( \widehat{q }_{0},\widehat{q}_{1}\right)&= \int \limits _{\widehat{q}_{1}}^{\overline{q_{1}}}\frac{ dW^{d}\left( r_{0}\left( q_{1}\right) , q_{1}\right) }{dq_{1}}dq_{1} \nonumber \\&= \int \limits _{ \widehat{q}_{1}}^{\overline{q_{1}}}\frac{\partial W^{d}\left( r_{0}\left( q_{1}\right) , q_{1}\right) }{\partial q_{1}}dq_{1} \end{aligned}$$

  (13)

  where the latter equality derives from an application of the Envelope Theorem. Hence,

  $$\begin{aligned} \frac{\partial W^{d}\left( r_{0}\left( q_{1}\right) , q_{1}\right) }{\partial q_{1}}&= P\left( r_{0}\left( q_{1}\right) +q_{1}\right) -C_{1}^{\prime }\left( q_{1}\right) \\&\ge P\left( r_{0}\left( q_{1}\right) +q_{1}\right) -C_{1}^{\prime }\left( q_{1}\right) \\&+\left[ 1+r_{0}^{\prime }\left( q_{1}\right) \right] q_{1}P^{\prime }\left( r_{0}\left( q_{1}\right) +q_{1}\right) \\&= \frac{d\varPi _{1}\left( r_{0}\left( q_{1}\right) , q_{1}\right) }{dq_{1}} \end{aligned}$$

  where the inequality follows from the fact that $-1\le r_{0}^{\prime }\left( q_{1}\right) \le 0,\forall q_{1}\ge 0$. Then:

  $$\begin{aligned} \int \limits _{\widehat{q}_{1}}^{\overline{q_{1}}}\frac{dW^{d}\left( r_{0}\left( q_{1}\right) ,q_{1}\right) }{dq_{1}}dq_{1}&\ge \int \limits _{\widehat{q}_{1}}^{ \overline{q_{1}}}\frac{d\varPi _{1}\left( r_{0}\left( q_{1}\right) ,q_{1}\right) }{dq_{1}}dq_{1} \\&= \varPi _{1}\left( \underline{q_{0}},\overline{q_{1} }\right) -\varPi _{1}\left( \widehat{q}_{0},\widehat{q}_{1}\right) >0 \end{aligned}$$

  where the latter inequality comes from the result in point (i).

This completes the proof. $\square$

Lemma 8

Consider the case of a domestic private firm. Assume that the game is of strict strategic substitutes, and that the public firm’s cost function is convex. Then, the output of the private firm (public firm) in any Stackelberg game with both orders of moves is strictly larger (strictly smaller) than in the Pareto-dominant mixed duopoly Cournot equilibrium.

Proof

We distinguish the two cases of (i) private leadership, and (ii) public leadership.

• (i) The case of private leadership has been already discussed in Lemmas 5-6 where it is shown that $q_{1}^{l}>\overline{q_{1}}$ and $q_{0}^{f}<\underline{q_{0}}$ always, and $\left( \underline{q_{0}},\overline{q_{1}}\right) \not \in S_{1}$.

• (ii) The case of public leadership is instead different since $W^{d}$ is not always increasing in $q_{1}$. We need therefore a different proof. Suppose by contradiction that $q_{1}^{l}<\overline{q_{1}}$. First observe that, since $r_{0}\left( q_{1}^{f}\right)$ is a best response,

  $$\begin{aligned} W^{d}\left( q_{0}^{l},q_{1}^{f}\right) \le W^{d}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) . \end{aligned}$$

  (14)

  Moreover, given that both $\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right)$ and $\left( \underline{q_{0}},\overline{q_{1}}\right)$ lie on $GR\,r_{0}\left( \cdot \right)$, we can apply the envelope theorem as in Eq. (13) and obtain the following relation between welfare and profits

  $$\begin{aligned} W^{d}\left( \underline{q_{0}},\overline{q_{1}}\right) -W^{d}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) \ge \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) -\varPi _{1}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) >0 \nonumber \\ \end{aligned}$$

  (15)

  (15) comes from the fact that

  $$\begin{aligned} \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge \varPi _{1}\left( \underline{q_{0}},q_{1}^{f}\right) >\varPi _{1}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) \end{aligned}$$

  where the first inequality comes from $\overline{q_{1}}$ being a best response to $\underline{q_{0}}$, and the second is due to the fact that $\varPi _{1}\left( \cdot \right)$ is strictly decreasing in $q_{0}$ and $r_{0}\left( q_{1}^{f}\right) >\underline{q_{0}}$.²⁴ So, combining inequalities (14) and (15), whenever $q_{1}^{f}<\overline{q_{1}}$, the following relation holds

  $$\begin{aligned} W^{d}\left( q_{0}^{l},q_{1}^{f}\right) \le W^{d}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) <W^{d}\left( \underline{q_{0}},\overline{ q_{1}}\right) , \end{aligned}$$

  which contradicts the nature of Stackelberg equilibrium. Therefore, it must be that $q_{1}^{f}\ge \overline{q_{1}}$ and $q_{0}^{l}\le \underline{ q_{0}}$. Moreover, we can apply the result in point $(ii)$ of the proof of Lemma 6 to show that $\left( \underline{q_{0}},\overline{q_{1}} \right) \not \in S_{0}$ and to conclude that the inequalities are strict, i.e., $q_{1}^{f}>\overline{q_{1}}$ and $q_{0}^{l}<\underline{q_{0}}$.

This completes the proof. $\square$

Proof of Theorem 4

To prove this result we need to show that both firms prefer the follower outcome to any simultaneous play solution whenever the Pareto dominant Cournot equilibrium $\left( \underline{q_{0}},\overline{q_{1}}\right)$ is interior.

• (i) By point (i) of Lemma 8 we know that any $\left( q_{0}^{f},q_{1}^{l}\right) \in S_{1}$ is such that $q_{0}^{f}< \underline{q_{0}}$ and $q_{1}^{l}>\overline{q_{1}}$. Moreover, from the definitions of Cournot and Stackelberg equilibria, both $\left( q_{0}^{f},q_{1}^{l}\right)$ and $\left( \underline{q_{0}},\overline{q_{1}} \right)$ lay on $GR\,r_{0}\left( q_{1}\right)$. Then, as in proof of Lemma 7, we can apply the Envelope Theorem and obtain the following relation between welfare and profits:

  $$\begin{aligned} W^{d}\left( q_{0}^{f},q_{1}^{l}\right) -W^{d}\left( \underline{q_{0}}, \overline{q_{1}}\right) \ge \varPi _{1}\left( q_{0}^{f},q_{1}^{l}\right) -\varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) >0 \end{aligned}$$

  (16)

  where the latter inequality comes from the property of Stackelberg equilibria, and it is strict by the result in point $(i)$ of the proof of Lemma 6.

• (ii) By point (ii) of Lemma 8, we know that any $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ is such that $q_{0}^{l}< \underline{q_{0}}$ and $q_{1}^{f}>\overline{q_{1}}$. Then it follows that

  $$\begin{aligned} \varPi _{1}\left( q_{0}^{l},q_{1}^{f}\right) \ge \varPi _{1}\left( q_{0}^{l}, \overline{q_{1}}\right) >\varPi _{1}\left( \underline{q_{0}},\overline{q_{1}} \right) \end{aligned}$$

  (17)

  where the first inequality comes from the fact that $q_{1}^{f}$ is best response to $q_{0}^{l}$ and the second is due to the private firm’s objective being strictly decreasing in $q_{0}$.

The conditions of point (ii) of Proposition 1 hold and both Stackelberg equilibria are SPE of the endogenous timing game. $\square$

Proof of Theorem 5

Here $r_{1}\left( q_{0}\right)$ is strictly increasing and $r_{0}\left( q_{1}\right)$ is strictly decreasing, when interior. Hence, the mixed duopoly Cournot equilibrium is unique (call it $\left( \underline{q_{0}}, \overline{q_{1}}\right)$). As in the proof of Theorem 3, we now prove that: (i) the public firm is better off in the private leadership than in the mixed-duopoly Cournot equilibrium; while (ii) the private firm’s payoff is strictly larger in the Cournot than in the public leadership equilibrium.

• (i) For the private leadership equilibrium, since $r_{0}$ is strictly decreasing in the interior, we can apply the same analysis as in point $(i)$ in Theorem 4 to show that from inequality (16) it follows that

  $$\begin{aligned} W^{d}\left( q_{0}^{f},q_{1}^{l}\right) >W^{d}\left( \underline{q_{0}}, \overline{q_{1}}\right) \end{aligned}$$

• (ii) We now analyze the public firm’s leadership equilibrium. We first show the public leadership equilibrium $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ is such that $q_{0}^{l}>\underline{q_{0} }$ and $q_{1}^{f}>\overline{q_{1}}$. Suppose by contradiction that $q_{0}^{l}<\underline{q_{0}}$. First observe that

  $$\begin{aligned} W^{d}\left( q_{0}^{l},q_{1}^{f}\right) \le W^{d}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) \end{aligned}$$

  (18)

  because $r_{0}\left( q_{1}^{f}\right)$ is a best response. Moreover, given that both $\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right)$ and $\left( \underline{q_{0}},\overline{q_{1}}\right)$ lay on $GR\,r_{0}\left( .\right)$ with $q_{1}^{F}<\overline{q_{1}}$, we can apply the envelope theorem as in Eq. (13) to show that

  $$\begin{aligned} W^{d}\left( \underline{q_{0}},\overline{q_{1}}\right) -W^{d}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) \ge \varPi _{1}\left( \underline{ q_{0}},\overline{q_{1}}\right) -\varPi _{1}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) >0 \nonumber \\ \end{aligned}$$

  (19)

  The latter inequality comes from the fact that

  $$\begin{aligned} \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge \varPi _{1}\left( \underline{q_{0}},q_{1}^{f}\right) >\varPi _{1}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) \end{aligned}$$

  where the first inequality comes from $\overline{q_{1}}$ being a best response to $\underline{q_{0}}$, and the second is due to the fact that $\varPi _{1}\left( \cdot \right)$ is strictly decreasing in $q_{0}$ and $r_{0}\left( q_{1}^{f}\right) >\underline{q_{0}}$. Considering the two inequalities (18) and (19), it follows that

  $$\begin{aligned} W^{d}\left( q_{0}^{l},q_{1}^{f}\right) \le W^{d}\left( r_{0}\left( q_{1}^{f}\right) ,q_{1}^{f}\right) <W^{d}\left( \underline{q_{0}},\overline{ q_{1}}\right) \end{aligned}$$

  which contradicts the nature of Stackelberg equilibrium. Therefore it must be that $q_{0}^{l}\ge \underline{q_{0}}$ and $q_{1}^{f}\ge \overline{q_{1}}$. Moreover, we can apply the result in point $(ii)$ of the proof of Lemma 6 to show that $\left( \underline{q_{0}},\overline{q_{1} }\right) \not \in S_{0}$ and to conclude that the inequalities are strict. So, since the Stackelberg equilibrium $\left( q_{0}^{l},q_{1}^{f}\right) \in S_{0}$ is such that $q_{0}^{l}>\underline{q_{0}}$ and $q_{1}^{f}>\overline{ q_{1}}$, then

  $$\begin{aligned} \varPi _{1}\left( \underline{q_{0}},\overline{q_{1}}\right) \ge \varPi _{1}\left( \underline{q_{0}},q_{1}^{f}\right) >\varPi _{1}\left( q_{0}^{l},q_{1}^{f}\right) \end{aligned}$$

  where the first inequality follows from the property of Cournot equilibria and the second from the fact that $\varPi _{1}$ is strictly decreasing in $q_{0}$. Then, the private firm strictly prefers Cournot to public leadership equilibria. Therefore the conditions of point (iii) of Proposition 1 hold and the private leadership equilibrium is the unique SPE of the endogenous timing game. $\square$