Abstract
This study investigates an endogenous R&D timing game between duopoly firms which undertake cost-reducing R&D investments simultaneously or sequentially, and then play Cournot output competition. We examine and compare equilibrium outcomes in private and mixed markets, respectively, and find that spillovers rate critically affects contrasting results. We show that a simultaneous-move (sequential-move) appears in a private duopoly if the spillovers rate is low (high) while a sequential-move appears in a mixed duopoly irrespective of spillovers. We also show that public leadership is the only equilibrium if the spillovers rate is intermediate and its resulting welfare is the highest. This suggests that the implementation of privatization policy transforms a public leader to a private competitor, which can decrease the social welfare.
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Notes
In an endogenous timing game in mixed market, Bárcena-Ruiz (2007), Lu (2006), Lu and Poddar (2009), and Heywood and Ye (2009) examined different contexts. For recent analysis, see Bárcena-Ruiz and Garzón (2010), Tomaru and Kiyono (2010), Balogh and Tasnádi (2012), Amir and De Feo (2014), Matsumura and Ogawa (2010, 2014, 2017), Naya (2015), Din and Sun (2016), Lee and Xu (2018), Garcia et al. (2019), and Cho et al. (2019) among others.
Knowledge and benefits obtained by a firm from its R&D activities might leak out to other firms. Thus, the firm cannot appropriate all the fruit of its R&D activities. Since d’Aspremont and Jacquemin (1988) and Suzumura (1992), the economics of R&D spillovers has been one of the most active issues in industrial economics over the last two decades. For example, Katsoulacos and Ulph (1998), Poyago-Theotoky (1995, 1999), Amir et al. (2000), and Baranes and Tropeano (2003) constructed a theoretical model of endogenous R&D spillovers.
The model with linear demand and quadratic cost functions is a standard formulation, which makes it possible to obtain a closed-form solution for each of the game and allows for a comparison of the equilibrium outcomes. It is also popularly used in the literature of mixed oligopolies since it can rule out an uninteresting corner solution case of a public monopoly. See, for example, Gil-Molto et al. (Gil-Molto et al. 2011; Gil-Molto et al. 2020), Kesavayuth and Zikos (2013), Lee et al. (2017), and Leal et al. (Leal et al. 2018, Leal et al. 2021).
Note also that the difference of marginal cost between the two firms is 2(qi − qj) − (1 − β)(xi − xj). It implies that the distribution of production costs across the firms is efficient only under the symmetric equilibrium where qi = qj and xi = xj. For more economic rationale behind this formulation, see Matsumura and Okamura (2015) and Kim et al. (2019).
Due to the linear-quadratic structure of the model, it is easy to show that all the second-order conditions of the equilibrium outcomes in this study are satisfied.
The proofs of lemmas and propositions are provided in Appendix.
If we consider the first-best outcome where the welfare is directly maximized by R&D (xFB) and output production (qFB), we have \( {x}_0^{CM}\frac{>}{<}{x}^{FB} \) if \( \beta \frac{<}{>}0.213 \) while \( {x}^{FB}>{x}_1^{CM} \) for any β ∈ [0, 1]. Thus, if the spillovers rate is low, the public firm chooses over-investment while the private firm always chooses under-investment.
Note that \( {q}_0^{CM}\frac{>}{<}{q}^{FB} \) if \( \beta \frac{<}{>}0.576 \) while \( {q}^{FB}>{q}_1^{CM} \) for any β ∈ [0, 1]. Thus, if the spillovers rate is not so high, the public firm chooses over-production while the private firm always chooses under-production.
Note that \( {x}_0^{LM}\frac{>}{<}{x}^{FB} \) if \( \beta \frac{<}{>}0.181 \) while \( {x}^{FB}>{x}_1^{LM} \). Thus, if the spillovers rate is low, the public leader chooses over-investment while the private follower always chooses under-investment.
Note that \( {q}_0^{LM}\frac{>}{<}{q}^{FB} \) if \( \beta \frac{<}{>}0.59 \) while \( {q}^{FB}>{q}_1^{LM} \). Thus, if the spillovers rate is not so high, the public leader chooses over-production while the private follower always chooses under-production.
Note that \( {x}_0^{FM}\frac{>}{<}{x}^{FB} \) if \( \beta \frac{<}{>}0.213 \) while \( {x}^{FB}>{x}_1^{LM} \). Thus, if the spillovers rate is low, the public follower chooses over-investment while the private leader always chooses under-investment.
Note that \( {q}_0^{FM}\frac{>}{<}{q}^{FB} \) if \( \beta \frac{<}{>}0.583 \) while \( {q}^{FB}>{q}_1^{FM} \). Thus, if the spillovers rate is not so high, the public follower chooses over-production while the private leader always chooses under-production.
From the strategic relation of R&D decisions, it is noted that the private firm might reduce its R&D when the public firm chooses over-investment while the public firm might reduce its R&D when the private firm chooses under-investment, especially when the spillovers rate is intermediate, 0.165 < β < 0.333. Thus, when the public firm is able to control the R&D decisions under the public leadership, it can reduce its over-investment, compared to the simultaneous-move game.
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APPENDIX
APPENDIX
Proof of Lemma 1:
Comparing total R&D and total outputs in (8), (11) and (9), (12), respectively, we obtain XCP − XLP ≤ 0 and QCP − QLP ≤ 0 where the equality satisfies if β = 0.25
Proof of Lemma 2:
Comparing the profits of firms in (10) and (13), we can obtain (i) \( {\pi}_0^{LP}-{\pi}_i^{CP}\ge 0 \) where the equality satisfies if β = 0.25, (ii) \( {\pi}_1^{LP}-{\pi}_i^{CP}\ \frac{>}{<}0\ \mathrm{if}\ \beta \frac{>}{<}0.25 \), and (iii) \( {\pi}_0^{LP}-{\pi}_1^{LP}\ \frac{>}{<}0\ \mathrm{if}\ \beta \frac{<}{>}0.25. \)
Proof of Lemma 3:
Comparing welfares in (10) and (14), we obtain WCP − WLP ≤ 0 where the equality satisfies if β = 0.25.
Proof of Lemma 4:
First, comparing total R&D derived in (20), (23), and (26), we obtain (i) \( {X}^{LM}-{X}^{CM}\frac{>}{<}0\kern0.5em \mathrm{if}\kern0.5em \beta \frac{>}{<}0.333; \) (ii) XFM − XCM < 0 if 0.165 < β < 0.333, but the reverse holds otherwise; (iii) XLM − XFM > 0 if 0.333 < β < 0.558, but the reverse holds otherwise.
Second, comparing total outputs in (21), (24), and (27), we obtain (i) \( {Q}^{LM}-{Q}^{CM}\frac{>}{<}0\kern0.5em \mathrm{if}\kern0.5em \beta \frac{>}{<}0.333; \) (ii) QFM − QCM < 0 if 0.165 < β < 0.333, but the reverse holds otherwise; (iii) QLM − QFM > 0 if 0.333 < β < 0.611, but the reverse holds otherwise.
Proof of Lemma 5:
Comparing the profits of firms in (22), (25), and (28), we obtain (i) \( {\pi}_1^{LM}-{\pi}_1^{CM}\ge 0, \) (ii) \( {\pi}_1^{FM}-{\pi}_1^{CM}\ge 0, \) and (iii) \( {\pi}_1^{LM}-{\pi}_1^{FM}\ge 0 \)where the equality satisfies if β = 0.333.
Proof of Lemma 6:
Comparing welfares in (22), (25), and (28), we obtain (i) WLM − WCM ≥ 0 where the equality satisfies if β = 0.333; (ii) WFM − WCM < 0 if 0.165 < β < 0.333, but the reverse holds otherwise; (iii) WFM − WLM < 0 if 0.163 < β < 0.333, but the reverse holds otherwise.
Proof of Proposition 5:
Comparing social welfares between private and mixed duopoly markets, we obtain WFM − WLP > 0 and WLM − WLP > 0 for any β ∈ [0, 1]. Then, from Lemmas 3 and 6, we have WCM > WCP and min{WFM, WLM} > WLP ≥ WCP where the equality satisfies if β = 0.25. Therefore, we have the following: (i) WCP < WLM < WFM if 0 < β < 0.163; (ii) WCP < WFM < WLM if 0.163 < β < 0.25, (iii) WLP < WLM if 0.25 < β < 0.333; (iv) WLP < WLM < WFM if 0.333 < β < 1.
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Lee, SH., Muminov, T.K. Endogenous Timing of R&D Decisions and Privatization Policy with Research Spillovers. J Ind Compet Trade 21, 505–525 (2021). https://doi.org/10.1007/s10842-021-00365-5
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DOI: https://doi.org/10.1007/s10842-021-00365-5