Relative profit maximization and Bertrand equilibrium with quadratic cost functions

Satoh, Atsuhiro; Tanaka, Yasuhito

doi:10.17811/ebl.2.3.2013.134-139

Cited by 9 publications

(7 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Satoh and Tanaka (2013), we see a similar result in the case of linear demand functions and quadratic cost functions. In this study, we extend that result to a case of general demand functions and convex cost functions.…”

Section: Introductionsupporting

confidence: 74%

Relative Profit Maximization and Bertrand Equilibrium with Convex Cost Functions

Satoh

Tanaka

2014

Economics

Self Cite

View full text Add to dashboard Cite

This study derives pure strategy Bertrand equilibria in a duopoly in which two firms produce a homogeneous good with convex cost functions and seek to maximize the weighted sum of their absolute and relative profits. The study shows that there exists a range of equilibrium prices in duopolistic equilibria. This range of equilibrium prices is narrower and lower than the range of equilibrium prices in duopolistic equilibria under pure absolute profit maximization. Moreover, the larger the weight on the relative profit, the narrower and lower the range of equilibrium prices. In this sense, relative profit maximization is more aggressive than absolute profit maximization. JEL D43 L13

show abstract

Section: Introductionsupporting

confidence: 74%

Relative Profit Maximization and Bertrand Equilibrium with Convex Cost Functions

Satoh

Tanaka

2014

Economics

Self Cite

View full text Add to dashboard Cite

show abstract

“…4 About relative profit maximization under imperfect competition, please see Matsumura, Matsushima and Cato(2013), Satoh and Tanaka (2013), Satoh and Tanaka (2014a), Satoh and Tanaka (2014b), Tanaka (2013a), Tanaka (2013b) and Vega-Redondo(1997 dx p x dp dx x x dx x dx p x dp…”

Section: B a Bmentioning

confidence: 99%

Maximin and Minimax Strategies in Two-Players Game with Two Strategic Variables

Satoh

Tanaka

2018

Int. Game Theory Rev.

Self Cite

View full text Add to dashboard Cite

We examine maximin and minimax strategies for players in two-players game with two strategic variables x and p . We consider two patterns of game; one is the x -game in which strategic variables of players are x 's, and the other is the p -game in which strategic variables of players are p 's. We call two players Players A and B, and will show that the maximin strategy and the minimax strategy in the x -game, and the maximin strategy and the minimax strategy in the p -game are all equivalent for each player. However, the maximin strategy for Player A and that for Player B are not necessarily equivalent, and they are not necessarily equivalent to their Nash equilibrium strategies in the x -game nor the p -game. But, in a special case, where the objective function of Player B is the opposite of the objective function of Player A, the maximin strategy for Player A and that for Player B are equivalent, and they constitute the Nash equilibrium both in the x -game and the p -game.

show abstract

“…The objective of this note is to show the key differences in the outcome of Bertrand competition under different cost functions of firms. In particular, we consider the homogenous duopoly competition when the cost functions are quadratic, and compare it with the results of the standard Bertrand models with demand-determined output discussed in the previous literature, summarized by, for example, Dastidar [1995], or Satoh and Tanaka [2013].…”

Section: Introductionmentioning

confidence: 99%

A Note on Bertrand Competition under Quadratic Cost Functions

Prokop

Ramsza

Wiśnicki

2015

Gospodarka Narodowa

View full text Add to dashboard Cite

The authors focus on a model of competition known in economics as Bertrand competition. They investigate how the outcome of Bertrand competition changes when linear cost functions are replaced by quadratic functions in the model. Named after French mathematician Joseph Louis Francois Bertrand (1822-1900), Bertrand competition describes interactions among firms and a market situation in which firms make their output and pricing decisions based on the assumption that their competitors will not change their own prices. Prokop, Ramsza and Wiśnicki show that the introduction of quadratic cost functions in the model leads to a qualitative change in competition between firms. In this case, the standard assumption that a firm with a lower price is interested in taking over the entire market is not only unrealistic (even in the absence of capacity constraints), but also irrational from the profit-maximization viewpoint, the authors say. Therefore the results of previous studies are misleading, according to Prokop, Ramsza and Wiśnicki. They argue that the so-called Bertrand duopoly model should be adjusted to better capture the behavior of firms under quadratic cost functions. The authors relax the assumption that the output of firms is determined by existing demand. They offer a modified Bertrand duopoly model in which firms competing in prices are free to choose their level of production depending on the market demand for their products at specific prices. "Under the modified assumptions, the static price competition game of identical duopolists has no symmetric equilibrium in pure strategies," the authors conclude. Their research shows that, under quadratic cost functions, it is possible to expect price fluctuations on oligopolistic markets rather than a stable equilibrium situation described by the standard model of price competition.

show abstract

Relative profit maximization and Bertrand equilibrium with quadratic cost functions

Cited by 9 publications

References 9 publications

Relative Profit Maximization and Bertrand Equilibrium with Convex Cost Functions

Relative Profit Maximization and Bertrand Equilibrium with Convex Cost Functions

Maximin and Minimax Strategies in Two-Players Game with Two Strategic Variables

A Note on Bertrand Competition under Quadratic Cost Functions

Contact Info

Product

Resources

About