Skip to main content
Log in

Comparing Cournot and Bertrand Equilibria in the Presence of Spatial Barriers and R&D

  • Published:
Review of Industrial Organization Aims and scope Submit manuscript

Abstract

We compare the equilibria under Bertrand and Cournot competition in the spatial barbell model where spatial barriers and process R&D are involved. We show that when the market becomes more competitive by switching from Cournot to Bertrand competition, R&D investment may increase (decrease) depending upon a low (high) transport rate. Next, we find that under Cournot competition total output, consumer surplus, and welfare are higher, but profit is lower than is true for Bertrand competition, when the transport rate is high, which overturns the traditional result.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Theoretically, Schumpeter (1943) argues that firms are more innovative in a weaker competitive market, while Arrow (1962) reaches the opposite conclusion. The empirical evidence on the relationship between competition and innovation is also mixed. See, for example, Aghion et al. (2005) and Tang (2006).

  2. Some studies—such as Delbono and Denicolo (1990), Mukherjee (2011), and Chang et al. (2017)—also compare the equilibria between Cournot and Bertrand competition.

  3. If the firms engage in Bertrand competition in each market, then each firm has the opportunity to price discriminate between the two markets by charging a limit price that is slightly lower than the rival’s marginal cost plus transport cost in its advantageous market, and a limit price that equals its own marginal cost plus transport cost in the remote market.

  4. By equating the equilibrium price with the monopoly price, we can obtain the cap on the transport rate in (5). Note that the condition—\( \left| {c_{A} - c_{B} } \right| \le t[\left( {1 - x_{B}^{U} } \right) - x_{A}^{U} ] \)—is needed in deriving (5) to ensure that the high-cost firm can survive. Intuitively, as the monopoly profit is the largest profit that the firms can earn, the firms will keep charging the monopoly price to earn this largest profit even though they are capable of charging a limit price that is higher than the monopoly price when the transport rate is higher than \( \overline{t} \). Accordingly, we define \( \overline{t} \) as the cap on the transport rate under Bertrand competition, because all of the results for those transport rates that are higher than \( \overline{t} \) are the same as the result at \( \overline{t} \).

  5. Note that the total output and then the price in each market could differ across the two markets, if the markets and the firms are asymmetric. However, they are identical across the two markets in equilibrium when the markets and the firms are symmetric.

  6. The second-order and the stability conditions are all fulfilled.

  7. From the cap of t in (5), we can obtain this inequality.

  8. The restriction, \( \gamma > 8/3 \), is the stability condition under Cournot competition, which can ensure that both the second-order condition under Cournot competition and the stability condition under Bertrand competition are satisfied. The proof can be provided by the authors upon request.

  9. By substituting the equilibrium values of \( \varepsilon_{i}^{{U^{*} }} \) and (x U*A , x U*B ) = (0, 0), which are solved in stages 1 and 2, into \( \bar{t} \) in (5), we can rewrite \( \bar{t} \) in a reduced form denoted by \( \overline{\overline{t}} = \gamma \left( {1 - c} \right)/\left( {2\gamma - 1} \right) \), which contains exogenous variables only. \( \overline{\overline{t}} \) is denoted as a threshold of the transport rate, which will lead the limit price to be equal to the monopoly price in the equilibrium under Bertrand competition. The restriction t < \( \overline{\overline{t}} \) corresponds to the case where the monopolist will never charge a limit price that is higher than the monopoly price in the equilibrium under Bertrand competition.

  10. Provided that the firms’ locations are (x U*A , x U*B ) = (x C*A , x C*B ) = (0, 0) under Bertrand competition and Cournot competition, respectively, we can obtain from (2), (3), and (7) that \( q_{ki}^{U} = 1 - t - c + \varepsilon_{i}^{{U^{*} }} \), and \( q_{Li}^{C} + q_{Ri}^{C} = \left[ {2\left( {1 - c + \varepsilon_{i}^{{C^{*} }} } \right) - t} \right]/3 \), i = A, B. Thus, Bertrand total output is greater than Cournot total output for the same t and εi, where \( t < \overline{\overline{t}} \).

  11. By differentiating the equations in footnote 10 with respect to t, we can derive that \( \partial q_{ki}^{U} /\partial t = - 1 < \left( {4/3} \right)\left( {\partial \left( {q_{Li}^{C} + q_{Ri}^{C} } \right)/\partial t} \right) = - 4/9 \) for any given εi.

  12. The difference between Cournot and Bertrand R&D is increasing in the transport rate, which can be proved by differentiating (15) with respect to t as \( \partial \left( {\varepsilon_{i}^{{C^{*} }} - \varepsilon_{i}^{{U^{*} }} } \right)/\partial t = \left( {5\gamma - 4} \right)/\left[ {\left( {\gamma - 1} \right)\left( {9\gamma - 8} \right)} \right] > 0 \) where γ > 8/3.

  13. Note that the total profit is the sum of the two firms’ profits. As the markets are symmetric, the two firms’ profits are identical, regardless of the competition mode.

  14. Differentiating Bertrand firm i’s profit with respect to t yields \( \frac{{d\pi_{i}^{U} }}{dt} = \frac{{\partial \pi_{i}^{U} }}{\partial t} + \frac{{\partial \pi_{i}^{U} }}{{\partial \varepsilon_{j}^{U} }}\frac{{\partial \varepsilon_{j}^{U} }}{\partial t}, i \ne j, i,j = A,B. \) The first and second terms on the right-hand side of the above equation denote the direct and R&D effects, respectively.

  15. Differentiating Cournot firm i’s profit with respect to t yields \( \frac{{d\pi_{i}^{C} }}{dt} = \left[ {\frac{{\partial \pi_{Li}^{C} }}{\partial t} + \frac{{\partial \pi_{Ri}^{C} }}{\partial t}} \right] + \left[ {\frac{{\partial \pi_{Li}^{C} }}{{\partial \varepsilon_{j}^{C} }}\frac{{\partial \varepsilon_{j}^{C} }}{\partial t} + \frac{{\partial \pi_{Ri}^{C} }}{{\partial \varepsilon_{j}^{C} }}\frac{{\partial \varepsilon_{j}^{C} }}{\partial t}} \right], i \ne j, i,j = A,B. \) The first and second terms on the right-hand side of the above equation denote the direct and R&D effects, respectively.

  16. The total surplus is the sum of the areas beneath the demand curves and above the marginal production cost curves in the two markets.

  17. Recall that γ > 8/3. We find from footnote 9 and (15) that \( t_{0} < \overline{\overline{t}} \).

  18. Recall that γ > 8/3 and footnote 9. We can obtain that \( \overline{\overline{t}} - t_{1} = 2\left( {1 - c} \right)\left( {\gamma - 1} \right)/\left[ {\left( {2\gamma - 1} \right)\left( {6\gamma - 5} \right)} \right] > 0 \).

  19. Recall that γ > 8/3 and footnote 9. We can figure out that

    $$ \overline{\overline{t}} - t_{2} = \frac{{\gamma \left( {1 - c} \right)\left[ {\sqrt {\left( {9\gamma^{2} + 2\gamma - 4} \right)\left( {9\gamma - 8} \right)^{2} \left( {\gamma - 1} \right)^{2} } \left( {2\gamma - 1} \right) + H_{5} } \right]}}{{H_{1} \left( {2\gamma - 1} \right)}} > 0, $$

    where \( H_{5} = 54\gamma^{4} - 186\gamma^{3} + 242\gamma^{2} - 142\gamma + 32 > 0 \).

  20. By manipulating, we obtain from footnote 9 and (19) that

    $$ t_{3} - \overline{\overline{t}} = \frac{{\gamma (1 - c)\left\lfloor {\sqrt {(4\gamma - 3)(9\gamma - 8)} (2\gamma - 1) - (12\gamma^{2} - 15\gamma + 4)} \right\rfloor }}{{(20\gamma^{2} - 27\gamma + 8)(2\gamma - 1)}} .$$

    Based on the restriction that γ > 8/3, the denominator is positive. Moreover, through manipulations we can show that the numerator is negative. It follows that \( t_{3} - \overline{\overline{t}} < 0 \).

References

  • Aghion, P., Bloom, N., Blundell, R., Griffith, R., & Howitt, P. (2005). Competition and innovation: An inverted-U relationship. Quarterly Journal of Economics, 120(2), 701–728.

    Google Scholar 

  • Arrow, K. (1962). Economic welfare and the allocation of resources for invention. In R. Nelson (Ed.), The rate and direction of inventive activity (pp. 609–626). Princeton, NJ: Princeton University Press.

    Chapter  Google Scholar 

  • Chang, R. Y., Hwang, H., & Peng, C. H. (2017). Competition, product innovation and licensing. The B.E. Journal of Economic Analysis & Policy. https://doi.org/10.1515/bejeap-2016-0136.

    Article  Google Scholar 

  • d’Aspremont, C., Gabzewicz, J., & Thisse, J. F. (1979). On Hotelling’s stability in competition. Econometrica, 47(5), 1145–1150.

    Article  Google Scholar 

  • Delbono, F., & Denicolo, V. (1990). R&D investment in a symmetric and homogeneous oligopoly. International Journal of Industrial Organization, 8, 297–313.

    Article  Google Scholar 

  • Hwang, H., & Mai, C. C. (1990). Effects of spatial price discrimination on output, welfare, and location. American Economic Review, 80, 567–575.

    Google Scholar 

  • Liang, W. J., Hwang, H., & Mai, C. C. (2006). Spatial discrimination: Bertrand vs. Cournot with asymmetric demands. Regional Science and Urban Economics, 36(6), 790–802.

    Article  Google Scholar 

  • Lin, P., & Saggi, K. (2002). Product differentiation, process R&D, and the nature of market competition. European Economic Review, 46, 201–211.

    Article  Google Scholar 

  • Mukherjee, A. (2011). Competition, innovation and welfare. Manchester School, 79(6), 1045–1057.

    Article  Google Scholar 

  • Qiu, L. (1997). On the dynamic efficiency of Bertrand and Cournot equilibria. Journal of Economic Theory, 75, 213–229.

    Article  Google Scholar 

  • Schumpeter, J. (1943). Capitalism, socialism and democracy. London: Allen & Unwin.

    Google Scholar 

  • Singh, N., & Vives, X. (1984). Price and quantity competition in a differentiated duopoly. Rand Journal of Economics, 15, 546–554.

    Article  Google Scholar 

  • Sun, C. H., & Lai, F. C. (2014). Spatial price discrimination in a symmetric barbell model: Bertrand vs. Cournot. Papers in Regional Science, 93, 141–158.

    Article  Google Scholar 

  • Tang, J. (2006). Competition and innovation behavior. Research Policy, 35, 68–82.

    Article  Google Scholar 

  • Wang, K. C. A., Tseng, C. C., & Liang, W. J. (2016). Patent licensing in the presence of trade barriers. Japanese Economic Review, 67(3), 329–347.

    Article  Google Scholar 

Download references

Acknowledgements

We are indebted to the editor and two anonymous referees for inducing us to improve our exposition and for offering several suggestions leading to improvements in the substance of the paper. The financial support from the Ministry of Science and Technology of Taiwan (MOST 104-2410-H-259-003-MY2) is gratefully acknowledged. The usual disclaimer applies.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wen-Jung Liang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

By subtracting (13) from (11), we obtainFootnote 17:

$$ \varepsilon_{i}^{{U^{*} }} - \varepsilon_{i}^{{C^{*} }} = \frac{{\left[ {\gamma \left( {1 - c} \right) - t\left( {5\gamma - 4} \right)} \right]}}{{\left( {\gamma - 1} \right)\left( {9\gamma - 8} \right)}} > \left( < \right)0,\quad {\text{if}}\; t < t_{0} \left( {t_{0} < t < \overline{\overline{t}} } \right), $$
(15)

where \( t_{0} = \frac{{\gamma \left( {1 - c} \right)}}{5\gamma - 4} \).

Next, by substituting (1), (3), (11), and (x U*A , x U*B ) = (0, 0) into (2), and substituting (13) and (x C*A , x C*B ) = (0, 0) into (7), we deriveFootnote 18:

$$ Q^{{U^{*} }} - Q^{{C^{*} }} = \frac{{2\gamma \left[ {\left( {1 - c} \right)\left( {3\gamma - 2} \right) - t\left( {6\gamma - 5} \right)} \right]}}{{\left( {\gamma - 1} \right)\left( {9\gamma - 8} \right)}} > \left( < \right)0,\quad {\text{if}}\; t < t_{1} \left( {t_{1} < t < \overline{\overline{t}} } \right), $$
(16)

where \( t_{1} = \frac{{\left( {1 - c} \right)\left( {3\gamma - 2} \right)}}{6\gamma - 5} \).

By substituting (1)–(3), (11) and (x U*A , x U*B ) = (0, 0) into (4), and substituting (7), (13) and (x C*A , x C*B ) = (0, 0) into (6), we can obtainFootnote 19:

$$ \begin{aligned} \pi_{i}^{{U^{*} }} - \pi_{i}^{{C^{*} }} & = \frac{{ - \left[ {t^{2} H_{1} - t\left( {1 - c} \right)H_{2} + \left( {1 - c} \right)^{2} H_{3} } \right]}}{{2\left( {\gamma - 1} \right)^{2} \left( {9\gamma - 8} \right)^{2} }} < \left( > \right)0, \\ & \quad {\text{if}} t < t_{2} \left( {t_{2} < t < \overline{\overline{t}} } \right),\quad i = A,B, \\ \end{aligned} $$
(17)

where

$$ \begin{aligned} & t_{2} = \frac{{\gamma \left( {1 - c} \right)\left( {99\gamma^{3} - 212\gamma^{2} + 146\gamma - 32 - \sqrt {H_{4} } } \right)}}{{H_{1} }}, \\ & H_{1} = 252\gamma^{4} - 709\gamma^{3} + 746\gamma^{2} - 352\gamma + 64 > 0, \\ & H_{2} = 198\gamma^{4} - 424\gamma^{3} + 292\gamma^{2} - 64\gamma > 0,H_{3} = 36\gamma^{4} - 55\gamma^{3} + 20\gamma^{2} > 0, \\ & H_{4} = 729\gamma^{6} - 2592\gamma^{5} + 2961\gamma^{4} - 358\gamma^{3} - 1700\gamma^{2} + 1216\gamma - 256 > 0. \\ \end{aligned} $$

By using the total outputs under Bertrand and Cournot competition, we derive the difference in consumer surplus between Bertrand and Cournot competition as:

$$ \begin{aligned} CS^{{U^{*} }} - CS^{{C^{*} }} & = \frac{1}{2}\left[ {\left( {Q_{L}^{{U^{*} }} } \right)^{2} + \left( {Q_{R}^{{U^{*} }} } \right)^{2} } \right] - \frac{1}{2}\left[ {\left( {Q_{L}^{{C^{*} }} } \right)^{2} + \left( {Q_{R}^{{C^{*} }} } \right)^{2} } \right] \\ & = \frac{{\gamma^{2} \left[ {\left( {1 - c} \right)\left( {15\gamma - 14} \right) - t\left( {12\gamma - 11} \right)} \right]\left[ {\left( {1 - c} \right)\left( {3\gamma - 2} \right) - t\left( {6\gamma - 5} \right)} \right]}}{{\left( {\gamma - 1} \right)^{2} \left( {9\gamma - 8} \right)^{2} }} \\ & \quad \quad \quad > \left( < \right)0, \quad {\text{if}}\; t < t_{1} \left( {t_{1} < t < \overline{\overline{t}} } \right). \\ \end{aligned} $$
(18)

We find from (17) and (18) that the difference in welfare between Cournot and Bertrand welfare is as followsFootnote 20:

$$ \begin{aligned} SW^{{C^{*} }} - SW^{{U^{*} }} & = \frac{{t^{2} \left( {20\gamma^{2} - 27\gamma + 8} \right) - 8\gamma t\left( {1 - c} \right)\left( {\gamma - 1} \right) - \gamma^{2} \left( {1 - c} \right)^{2} }}{{\left( {\gamma - 1} \right)\left( {9\gamma - 8} \right)}} \\ & < \left( > \right)0, \quad if\; t < t_{3} \left( {t_{3} < t < \overline{\overline{t}} } \right), \\ \end{aligned} $$
(19)

where \( t_{3} = \frac{{\gamma \left( {1 - c} \right)\left[ {4\left( {\gamma - 1} \right) + \sqrt {36\gamma^{2} - 59\gamma + 24} } \right]}}{{20\gamma^{2} - 27\gamma + 8}} \).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, KC.A., Wang, YJ. & Liang, WJ. Comparing Cournot and Bertrand Equilibria in the Presence of Spatial Barriers and R&D. Rev Ind Organ 58, 475–491 (2021). https://doi.org/10.1007/s11151-020-09775-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11151-020-09775-x

Keywords

JEL Classification

Navigation