Abstract
Production to order and production in advance have been compared in many frameworks. In this paper we investigate a production in advance version of the capacity-constrained Bertrand-Edgeworth mixed duopoly game and determine the solution of the respective timing game. We show that a pure-strategy (subgame-perfect) Nash-equilibrium exists for all possible orderings of moves. It is pointed out that unlike the production-to-order case, the equilibrium of the timing game lies at simultaneous moves. An analysis of the public firm’s impact on social surplus is also carried out. All the results are compared with those of the production-to order version of the respective game and with those of the mixed duopoly timing games.
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Notes
- 1.
The PIA game is also frequently called the price-quantity game or briefly PQ-game.
- 2.
We call an oligopoly standard if all firms are profit maximizers, which basically means that they are privately owned.
- 3.
- 4.
From the mentioned papers only Zhu et al. (2014) considered sequential orders of moves. For more on standard duopoly leader-follower games we refer to Boyer and Moreaux (1987), Deneckere and Kovenock (1992) and Tasnádi (2003) in the Bertrand-Edgeworth framework. Furthermore, Din and Sun (2016) extended Zhu et al. (2014) to mixed duopolies.
- 5.
- 6.
Note that \(D_i^r\left( p_i^m(q_j),q_j\right) \le k_i\) since \(p_i^m(q_j)\ge P(k_i+q_j)\).
- 7.
The equation defining \(p_i^d(q_j)\) has a solution for any \(q_j\in [0,k_j]\) if, for instance, \(p_i^m(q_j)\ge \max \{p^c,c\}\), which will be the case in our analysis when we refer to \(p_i^d(q_j)\).
- 8.
This ensures for the case when the public firm moves first the existence of a subgame perfect Nash equilibrium in order to avoid the consideration of \(\varepsilon \)-equilibria implying a more difficult analysis without substantial gain.
- 9.
In particular, there exists a subset \(\left[ \overline{q},k_1\right] \) of H.
- 10.
Observe that by Lemma 1, the monotonicity of \(p_2^d(\cdot )\), \(q_2^m(k_1)<k_2\) and \(P(k_1)>c\), we have \(p_2^*\ge p_2^d(q_1)\ge p^d_2(k_1)>\max \{p^c,c\}\).
- 11.
We recall that \(\tilde{q}_i\) has been defined after \(p_i^d(q_j)\).
- 12.
In particular, if the private firm sets a price not greater than \(p_1^*\), we are not anymore in Case C; if \(q_2^*>\min \left\{ D_2^r(p_2^*,q_1^*),k_2\right\} \), then the private firm produces a superfluous amount; if \(q_2^*<\min \left\{ D_2^r(p_2^*,q_1^*),k_2\right\} \), then the private firm could still sell more than \(q_2^*\); and if \(q_2^*=D_2^r(p_2^*,q_1^*)\), then the private firm will choose a price-quantity pair maximizing profits with respect to its residual demand curve \(D_2^r(\cdot ,q_1^*)\).
- 13.
Because \(p_2^m(\cdot )\) is a decreasing function in \(q_1\).
- 14.
We speak about family, because \(p_1^*\) can vary within a given range.
- 15.
As mentioned earlier in the weak private firm case we have \(p^c>c\).
- 16.
Depending on the parameters, it can also occur that the public firm has zero output on the residual demand curve.
- 17.
Recall that \(q_1<D(c)\Leftrightarrow P(q_1)>c\). In addition, \(q_1>0\) implies \(c<p_2^d\left( q_1\right) <p_2^m\left( q_1\right) \).
- 18.
If \(k_2\le D(p_2^*)\), a price decrease cannot increase the private firm’s profit, and if \(k_2>D(p_2^*)\), \(q_1^*=0\).
- 19.
Observe that this also implies \(P(q_1^*)>c\).
- 20.
It can be verified that we have obtained all \(NE_1\) type equilibria.
- 21.
We note that here \(p_1^*<c\), still, it is of the public firms interest to produce a positive amount, as this action leads to a positive change in consumer surplus. This is the reason why there is no producer surplus indicated on Fig. 4.
- 22.
To be precise if (24) is satisfied with equality, then both mentioned types are best responses; however, as it can be verified in a SPNE only the former type can be selected.
- 23.
Note that the distribution of production between the two firms does not effect (24).
- 24.
The latter assumption can be explained by risk dominance.
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This research is granted by the Pallas Athéné Domus Spientiae Foundation Leading Researcher Program.
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Balogh, T.L., Tasnádi, A. (2020). Mixed Duopolies with Advance Production. In: Szidarovszky, F., Bischi, G. (eds) Games and Dynamics in Economics. Springer, Singapore. https://doi.org/10.1007/978-981-15-3623-6_9
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