Price Reaction

One way to illustrate the zero-price equilibrium is to consider the reaction functions in the case of non-intersecting reaction functions. Each price reaction function, R, has a positive slope. The positive slope arises because the rule of the non co-operative game is a matching rule: one player reduces price and the second player follows with a price reduction and this continues in sequence until the two players drive the price towards the zero-price equilibrium at (0,0).

Initially both players are content with the price arrangements. A fact-finder might observe no price movements. Then one player, believing that the other player will not react, changes price, but the other player does in fact follow the price lead. The player who had initially considered the reduction in price on the belief that the rival player would not react, is faced with what we term the Bertrand dilemma. It is the belief that the second player would not follow that triggers the initial price move.

With intersecting price reaction functions there is a lower bound on the price reductions. Each player will not reduce price below a critical lower bound that reflects the financial health of the player. However there is always the possibility that one player might be a fighting ship, that is, a player cross-subsidised by a deep-pocket parent signalling an intention to trigger a price war with the rival, should the need arise in order to protect market share of a premium brand. Fighting ships are known as fighting brands in may product markets: for example, the brand Luvs could be regarded as a fighting brand for P&G in the market for diapers, wherein the brand Pampers could be regarded as P&G’s premium branded product.

The intersection point, E, is the best both players can do in terms of price, given the likely reaction of their respective rival. Hence we could label the equilibrium as a Nash equilibrium. It is the best each player can do, but it does not necessarily mean that it is the profit maximum or equal prices equilibrium. In the payoff matrix discussed under the section ‘Game Theory’ the (1,1) outcome can be interpreted as a Nash equilibrium; however, the win-win outcome is (2,2) and that can only be achieved with an illegal strategy of deliberately colluding to fix price to ensure the (2,2) outcome obtains.

Illustration 1 = Price War

Case of non-intersecting reaction functions

Here we illustrate the reaction functions used in non co-operative game theory to simply define a zero-price equilibrium. The two reaction functions have a positive slope because we are assuming a match-match rule of sequencing moves in price. Firm A moves first, not expecting B to react, B does react, and then A reacts in sequence matching each other until the price goes to zero.

Illustration

Case of intersecting reaction functions

The intersecting reaction functions arise because the two players realise that eventually they could end up at a zero-price. Neither shareholder nor management would wish to see a zero-price outcome, so we illustrate a minimum price at the points of intersection with each axis, at A and B points. The point at which both reaction functions intersect, point E, is the Nash equilibrium, it is the best each can do, given the reaction of the other in this matching of prices (check GAME THEORY).

Key Points

To illustrate reaction functions and to focus on their respective slopes and draw the price sequences.

To identify a zero-price equilibrium and a Nash equilibrium.