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Review of Industrial Organization

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16-07-2020

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

Authors: Kuang-Cheng Andy Wang, Yi-Jie Wang, Wen-Jung Liang

Published in: Review of Industrial Organization | Issue 3/2021

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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.

previous article Payoff-Improving Competition: Games with Negative Externalities

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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).

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

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.

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} $.

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.

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

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

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.

By substituting the equilibrium values of $ \varepsilon_{i}^{{U^{*} }} $ and (x _A ^U* , x _B ^U* ) = (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.

Provided that the firms’ locations are (x _A ^U* , x _B ^U* ) = (x _A ^C* , x _B ^C* ) = (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 ε_i, where $ t < \overline{\overline{t}} $.

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 ε_i.

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.

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.

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.

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.

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

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

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 $.

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 $.

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 $.

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Title: Comparing Cournot and Bertrand Equilibria in the Presence of Spatial Barriers and R&D
Authors: Kuang-Cheng Andy Wang
Yi-Jie Wang
Wen-Jung Liang
Publication date: 16-07-2020
Publisher: Springer US
Published in: Review of Industrial Organization / Issue 3/2021
Print ISSN: 0889-938X
Electronic ISSN: 1573-7160
DOI: https://doi.org/10.1007/s11151-020-09775-x

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