Chapter 3 Oligopoly Oligopoly is an industry where there are relatively few sellers. The product may be standardized (steel) or differentiated (automobiles). The firms have a high degree of interdependence. When one firm considers changing its price or output level, it must make assumptions about the reactions of its rivals.. The Cournot Model (Auguste Cournot, a French economist) Each firm assumes that its rivals will continue producing at current levels of output. Assume a firm oligopoly: a duopoly. Two firms selling bottled water from an identical mineral spring. Each firm treats the other s quantity of output as a fixed number, one that will not correspond to its own production decisions. Suppose the total market demand curve for mineral water is given by P = a - b (Q + Q ), (equation ) a, b > 0 Q, Q are outputs from firms and, respectively. Assume the marginal cost of producing water is zero. (for convenience results would be similar if the marginal costs were positive) Profit maximization problem for firm (given the assumption that Q is fixed): P = (a bq ) bq (equation ) For any level of Q, we can use equation to derive a demand curve for firm. For instance, if Q = 0, then firm would have the entire market demand curve. As Q increases, the balance of the market demand curve for firm decreases. Graph 3-
With reference to the graph above, Firm s demand curve is the portion of the original demand curve to the right of the new vertical axis. The new vertical axis is defined by firm s output. Firm s MR curve varies with the level of Q. Since MC = 0, the profit-maximizing level of output for firm is that level at which MR equals zero. MR = (a bq ) bq (twice the slope of demand) Set MR = MC and solve for Q in terms of output of firm. When MC=0, Q a bq = (equation 3) b Equation 3 is called firm s reaction function. It tells how firm s quantity will react to the quantity level of firm. The model is symmetric i.e. firm s reaction function has the same structure: Q = a bq b Plot both reaction functions:
The reaction function of each duopolist gives its profit-maximizing output level as a function of the other firm s output level. When are the duopolists in a stable equilibrium? At the point of intersection of bother reaction functions. In equilibrium each produces a/3b units of output. At equilibrium, each maximizes profit given the output decisions of the other. The combined output of the industry is a/3b. The market price will be P = a b(a/3b) = a/3 With this price, each will have total revenues equal to (a/3) (a/3b) = a /9b With no costs, total revenue = profit. A numerical example: Cournot duopolists face a market demand curve given by P = 56 Q, where Q is total market demand. Each can produce output at a constant marginal cost of $0/unit. Graph their reaction functions and find the equilibrium price and quantity. 3
. The Bertrand Model (Joseph Bertrand, a French economist) In this model, each firm chooses its price taking the rival s price as given. Suppose the market demand and cost conditions are the same as in the Cournot model. Firm charges an initial price of P 0. Firm is left with three choices:. charge more than firm (sell nothing). charge the same as firm (split the market) 3. charge marginally less than firm (capture entire market) This model is perfectly symmetrical meaning that the option of selling at a marginally lower price will be the strategy both firms will choose. There cannot be a stable equilibrium in which each firms undersells the other. Once each firm has cuts it price to marginal cost, there is no further incentive to cut further. In the end, the firms will equally share the market and sell at marginal cost. Numerical example: If the market demand curve facing Bertrand duopolists is given by P = 0 Q and each has a constant marginal cost of $/unit, what will be the equilibrium price and quantity for each firm? 4
3. The Stackelberg Model (Heinrich von Stackelberg, a German economist) This model is applicable to an industry in which one firm sets its profitmaximizing level of output first, knowing that its rival will behave as a Cournot duopolist. Suppose firm knows that firm will treat firm s output as given. Recall firm s reaction function: Q = ( a bq ) / b. Firm can substitute firm s reaction function for Q in the equation for market demand: P = a b[q + (a bq )/b] = (a- bq )/ Since MC = 0 in the mineral spring example, firm s profit-maximizing level of output will correspond to MR = 0: Q = a / b Q = a / 4b P = a bq = a b( Q + Q ) = a / 4 Insert figure 3-4 p 406 Numerical example: A Stackelberg leader (firm ) and follower (firm ) face a market demand curve given by P= 56 Q. Each can produce output at a constant marginal cost of $0/unit. Find the equilibrium price and quantity. 5
What happens if both firms attempt to be Stackelberg leaders? Each will ignore its own reaction function and produce a/b.which will result in total industry output and price to the same as in the Bertrand model, the competitive outcome. For consumers, this would be a good outcome, but for the firms, it is the worst possible outcome. Comparing the outcomes of the three models Cournot model: both firms produce a higher quantity and at a lower price than a monopolist. Bertrand model: both firms behave as competitive firms. Stackelberg model: the leader makes twice the profit of the follower. The combined output of both firms is greater than that in the Cournot model and the market price is lower than that in the Cournot model. 4. Game Theory Game theorists produce models of interactive strategic behaviour, and apply them to economic decision situations. There are 4 elements that characterize all games: ) players ) strategies (all possible strategies for each player) 3) payoffs (associated with each combination of strategies) 4) decision-rules (for each player in choosing among alternative strategies) Features of game models: - Games can be modelled as cooperative or non-cooperative. - There can be -player or multi-player games. - The payoff matrix or function can be zero-sum or non-zero-sum. - Zero-sum implies that one person s gain is another s loss. 6
- Players can have perfect information or asymmetric information. - Asymmetric information means that some information is not known to some players. - Games can be played one time only or repeated a definite number of times, or repeated an indefinite number of times. - Players can make moves simultaneously or sequentially. - Games can differ in the extent of communication permitted among players, the costs of communication, the possibility of side payments, and the extent to which coalition-formation is possible. Game theory decision rules: Assumptions: The players in a game are required to be rational. In other words, all players make decisions that are in their own best interest. As well, it is assumed that players are capable of determining which strategy will result in the best outcome for them. All players know the outcome to all players resulting from all possible strategy combinations. Payoffs to others do not influence their choices. As the number of players and number of strategies increases, the assumptions are more difficult to satisfy. Consider the table 3- (p 4) containing payoff matrices for four simple games. Each game has two players, Ron and Colleen. Each pair of strategies results in a payoff to each player. Ron plays one of two row strategies (r or r ). Colleen plays one of two column strategies (c or c ). Each pair of strategies results in a payoff to each player. 7
Ron s payoff is π R. Colleen s payoff is π C. (payoff is measured in dollars) Rules: Each game is played once. There is no communication among the players. No side payments are permitted. Determine the optimal strategy for each player. A dominant strategy exists if it produces a best outcome for a player regardless of the strategy chosen by the other player. Ron s dominant strategy in game a) is row. Colleen s dominant strategy in game a) is column. In games b), c), and d), Ron does not have a dominant strategy, as his payoff is $ regardless of this strategy. His strategy does affect Colleen s payoff. If Ron behaves as an Altruistic Egoist, he will choose his strategy yielding the best payoff for Colleen. (he costlessly creates goodwill, may generate indebtedness) If Ron behaves as a Misanthropic Egoist, he will choose his strategy yielding the worst payoff for Colleen. (he costlessly weakens a competitor) If Ron behaves as a Semi-Altruistic Egoist, he will choose his strategy such that Colleen is better off, but not where her profits are greater than his. (he costlessly creates goodwill up until the point where it does weaken his own position) If Ron behaves as an Indifferent Egoist, he will randomly choose his strategy. Consider the games from Colleen s perspective. She has a dominant strategy in both a) and b). In d), having some insight into Ron s behaviour (altruistic, misanthropic, etc.) will help Colleen choose her strategy. If Colleen believes that Ron is an Indifferent Egoist, there is a 50% chance of choosing row and a 50% chance of choosing row. Which column will she choose? Her expected payoff =.5(3) +.5(-) = $.50 versus =.5() +.5 (-) = $0, she chooses column. 8
If she isn t confident about being able to predict Ron s behaviour, she may minimize her maximum possible loss by choosing column. Or she could want to maximize her expected payoff, by choosing column. For Colleen, there is no one right decision rule. What column should Colleen choose for game c)? The Minimax criterion as a decision rule: minimize the maximum loss. The Maximin criterion as a decision rule: maximize the minimum gain. A loss is a minimum gain, so the two criterions are equivalent. How does the minimax criterion work? If Ron s strategy is known, she will select her strategy to maximize her payoff. If Ron s strategy is unknown, she will select the strategy that will minimize her maximum possible loss. This strategy is very conservative.a prepare for the worst case decision rule. The Nash equilibrium is the combination of strategies in a game such that no player has an incentive to change strategies, given the strategy of its rival. Is there a Nash equilibrium in game a)?, b)?, c)?, d)? Duopolist Applications Consider two tobacco companies and their decision to advertise. When a firm advertises its product, the demand for that firm s product will increase for two reasons:. People who have never used the product will learn about it and some will buy it.. Some of the people who use another brand of the product will switch brands because of the advertising. 9
Firm Firm Do not advertise Advertise Do not advertise Π = 500 Π = 500 Π = 750 Π = 0 Advertise Π = 0 Π = 50 Π = 750 Π = 50 Again, we assume no communication between the two firms. Does firm have a dominant strategy? Does firm have a dominant strategy? What is the Nash equilibrium? If the no communication assumption is removed, will the firms strategies change? 0