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Oligopoly - Competition among the Few

- In an oligopoly there are very few sellers of the good.
- The product may be differentiated among the sellers (e.g. automobiles) or homogeneous (e.g. gasoline).
- Entry is often limited either by legal restrictions (e.g. banking in most of the world) or by a very large minimum efficient scale (e.g. overnight mail service) or by strategic behavior.
- Sill assuming complete and full information.

How Oligopolists Compete

- In an oligopoly
- firms know that there are only a few large competitors;
- competitors take account of the effects of their actions on the overall market.
- To predict the outcome of such a market, economists must model the interaction between firms and so often use game theory or game theoretic principles.

Three Basic Models

- Competition in quantities: Cournot-Nash equilibrium
- Competition in prices: Bertrand-Nash equilibrium
- Collusive oligopoly: Chamberlin notion of conscious parallelism
- It is very useful to know some basic game theory to understand these models as well as other oligopoly models.

Game Theory: Setup

- List of players: all the players are specified in advance.
- List of actions: all the actions each player can take.
- Rules of play: who moves and when.
- Information structure: who knows what and when.
- Payoffs: the amount each player gets for every possible combination of the the players’ actions.

A Classic Two Player, Two Action Game - The Prisoners’ Dilemma

Chris

- Roger’s best response function:
- If Chris lies, then Roger should confess (check out left column, 1st entries)
- If Chris confesses, then Roger should confess (right column, 1st entries)
- Confess is a dominant strategy for Roger
- Chris’s best response function:
- If Roger lies, then Chris should confess (see top row, 2nd entries)
- If Roger confesses, then Chris should confess (bottom row, 2nd entries)
- Confess is a dominant strategy for Chris

Lie

Confess

Lie

-1, -1

-6, 0

Roger

Confess

0, -6

-5,-5

A Classic Two Player, Two Action Game - The Prisoners’ Dilemma

Chris

- There is a single dominant strategy equilibrium:
- Rogers confesses and
- Chris confesses
- They both go to jail for 5 years
- Note: the game is played simultaneously and non-cooperatively!
- Ways to sustain the cooperative equilibrium (lie, lie)
- different payoff structures
- repeated play and trigger strategies

Lie

Confess

Lie

-1,-1

-6, 0

Roger

Confess

0, -6

-5,-5

Question: Will There Always Be A Dominant Strategy Equilibrium?

- Answer…NO!
- Then what?
- Look for Nash Equilibrium.

Nash Equilibrium

- Named after John Nash - a Nobel Prize winner in Economics.
- The Nash Non-cooperative Equilibrium of a game is a set of actions for all players that, when played simultaneously, have the property that no player can improve his payoff by playing a different action, given the actions the others are playing.
- Each player maximizes his or her payoff under the assumption that all other players will do likewise.

Another Example - The Price Game

Chris

- Roger’s best response function:
- If Chris goes low, then Roger should go low (check out left column, 1st entries)
- If Chris goes high, then Roger should high (right column, 1st entries)
- There is no dominant strategy for Roger
- Chris’s best response function:
- If Roger goes low, then Chris should go low (see top row, 2nd entries)
- If Roger goes high, then Chris should go high (bottom row, 2nd entries)
- There is no dominant strategy for Chris

Low

High

Low

20, 20

60, 0

Roger

High

0, 60

100, 100

Another Example - The Price Game

Chris

- Roger’s best response function:
- If Chris goes low, then Roger should go low
- If Chris goes high, then Roger should high
- Chris’s best response function:
- If Roger goes low, then Chris should go low
- If Roger goes high, then Chris should go high
- Two Nash Equilibria: (low, low) and (high, high)
- Respective Nash equilibrium payoffs: (20,20) and (100,100)
- Which equilibrium will prevail? Good question.

Low

High

Low

20, 20

60, 0

Roger

High

0, 60

100, 100

Another Example - The Simultaneous Entry Game

Roger - the entrant

- Get two Nash equilibria:
- (enter, accommodate) and (not enter, fight)

enter

not enter

Chris - the incumbent

fight

fight

accommodate

accommodate

(Roger = 0,Chris = 0)

(Roger = 2, Chris = 2)

(Roger =1,Chris = 5)

(Roger =1,Chris = 5)

Another Example - The Sequential Entry Game

Roger - the entrant

- Still get two Nash equilibria:
- (enter, accommodate) and (not enter, fight)
- Only one, however, is credible: (enter, accommodate)

enter

not enter

Chris - the incumbent

fight

fight

accommodate

accommodate

(Roger = 0,Chris = 0)

(Roger = 2, Chris = 2)

(Roger =1,Chris = 5)

(Roger = 1,Chris = 5)

Another Two Player, Two Action Example

- The game has two players 1 & 2.
- Player 1 can move “up” or “down” (actions).
- Player 2 can move “left” or “right” (actions).
- If player 1 moves “up” and player 2 moves “left” then player 1 gets $1 and player 2 gets $0 (payoffs).
- The table shows all possible action pairs and their associated payoffs.

Player 1’s Best Strategies

- If player 2 plays “right,” the best strategy (action) for player 1 is to play “up.”
- In this case player 1 will get a payoff of $1, underlined.

Player 2’s Best Strategies

- If player 1 plays “up” then player 2’s best strategy (action) is to play “right.”
- In this case, player 2 gets a payoff of $2, underlined.

Nash Equilibrium

- The table shows the best strategy (actions) for player 1 against both of player 2’s possible actions (underlined first numbers).
- The table also shows the best strategy (actions) for player 2 against both of player 1’s possible actions (underlined second numbers).
- Notice that both numbers are underlined in the cell “up,right.” This is the Nash Equilibrium.
- If player 1 plays “up” the best thing for player 2 to do is play “right” and vice versa.

A Non-cooperative Outcome (Cournot-Nash Duopoly - Competition in Quantities)

- Developed by AntoineAugustinCournot in 1838.
- In a two firm oligopoly (called a duopoly), if both firms set their output levels assuming that the other firm’s strategic choice variable (quantities in Cournot competition) is fixed, the equilibrium outcome is a Cournot Nash Non-cooperative Equilibrium. (Note: Cournot solved this oligopoly model many years before Nash invented the equilibrium definition we are using here).

Setup of the Duopoly Problem: Monopoly Outcome

- The table at the right shows the monopolist’s best choice for the simple market demand curve shown, assuming only whole quantities can be chosen.
- The monopolist maximizes profits at X=3, P=$14, with economic profits of $21.
- Assuming only whole quantities can be produced, the competitive equilibrium is X=6, P=$8, the last price at which economic profits are not negative (FC=$0 and MC=$7 for all X).

Duopoly Game: Competition in Quantities

- Suppose that there are two firms X and Y with identical total cost curves that are the same ones shown for the monopolist in the previous slide: total cost=$7Xi
- The payoff matrix above shows the economic profits of Firm X (left entry) and Firm Y (right entry) for each possible quantity supplied of 0 to 4 units.
- The payoff for a firm is determined by finding the price that prevails for the total quantity supplied (Firm X + Firm Y), then multiplying each quantity by this price and subtracting the firm’s total costs for that quantity.
- Note: demand price is PD=20-2X where X=XX + XY
- Example: Firm X supplies 3 and Firm Y supplies 1 - so X=4 and P=12
- Firm X’s payoff = (3 x 12) - 21 = 15
- Firm Y’s payoff = (1 x 12) - 7 = 5

Duopoly Game: Nash Equilibrium in Quantities

- The boxes marked in yellow are the best moves for Firm X given the indicated quantity supplied by Firm Y.
- The boxes marked in green are the best moves for Firm Y given the indicated quantity supplied by Firm X.
- The payoff for the cell (X supplies 2, Y supplies 2) is (10, 10). This cell is the Nash Non-cooperative Equilibrium for this game because it represents the best move for Firm X given that Firm Y chooses its best move and the best move for Firm Y given that Firm X chooses its best move.
- Duopoly outcome: Total quantity supplied = 2 + 2 = 4. Market price = $12. Total economic profits = $10 + $10 = $20.
- Monopoly outcome: Total quantity supplied = 3. Market price = $14. Total economic profits = $21.
- Competitive outcome: Total quantity supplied = 6. Market price = $8. Total economic profits = $6.

Properties of the Cournot-Nash Equilibrium for Duopoly

- When the duopolists compete in quantities, we can compare the outcome to both the monopoly and competitive outcomes.
- Each duopolist produces less than a monopolist in the same market but together they produce more than the monopolist and less than the amount two competitive firms would have produced with the same cost structure and demand curves.
- The sum of the economic profits of each duopolist is less than the economic profits of a monopoly in the same market.
- The market price is less than the one a monopolist would charge but more than the competitive price.
- Deadweight loss is less than for a monopoly in the same market but still positive, thus greater than the deadweight loss from a competitive market.

Duopoly Game: Competition in Prices (J. Bertrand 1883)

- Firm X and Y have the same cost structure and face the same market as in the previous example.
- Now, instead of playing a game in quantities, they play a game in prices allowing only the choices indicated.
- The payoff matrix above shows the economic profits of Firm X (left entry) and Firm Y (right entry) for each possible price chosen $8, $10, $12, $14, $16.
- If the two firms choose the same price they split the market in half; otherwise, the firm that chooses the lower price sells the market quantity and the other firm sells nothing.
- Example: Firm X charges $12 and Firm Y charges $12
- Market X = 4, both firms sell 2 units at $12 and have total costs of $14.
- Firm X payoff = Firm Y payoff = 2 x $12 - $14 = $10.
- Example: Firm X charges $10 and Firm Y charges $8.
- Market X = 6, Firm Y sells all 6 units, Firm X sells nothing.
- Firm X payoff = $0; Firm Y payoff = 6 x $8 - $42 = $6.

Duopoly Game: Bertrand-Nash Equilibrium in Prices

- The boxes marked in yellow are the best moves for Firm X given the indicated quantity supplied by Firm Y.
- The boxes marked in green are the best moves for Firm Y given the indicated quantity supplied by Firm X.
- The payoff for the cell (X charges $8, Y charges $8) is (3, 3) and the payoff for the cell (X charges $10, Y charges $10) is (7.5, 7.5). Both cells are the Nash Non-cooperative Equilibria for this game.
- Duopoly competition in prices in this market does not have a unique equilibrium (a common occurrence in game theory).
- This game predicts that the market price fluctuates between $8 and $10.
- This game predicts that the market quantity fluctuates between 4 and 6.
- It is not uncommon for the competition in quantities game to give different results from the competition in prices game.

Performance: Bertrand vs. Cournot

- When the duopolists compete in prices, we can compare the outcome to both the monopoly and competitive outcomes, but it can be more difficult to find an equilibrium.
- Classic results (when an equilibrium exists and is unique).
- N=1 then XBN = XSM and PBN= PSM
- N>1 then XBN = X* and PBN = P*
- Bertrand compared to Cournot.
- N=1 then XCN = XSM and PCN= PSM
- N>1 then X* > XCN > XSM and P*< PCN < PSM
- N gets large enough, XCN = X* and PCN=P*
- Results have different implications for anti-trust action.
- Should MCI be able to merge with Sprint? N goes from 3 to 2.
- Should Coke be allowed to merge with Dr. Pepper? Should Pepsi be allowed to merge with 7-Up?
- Good questions.

A Cooperative Outcome (Collusion)

- The duopolists can do better than the Nash Non-cooperative Equilibrium.
- Because the equilibrium is non-cooperative, we have ruled out the possibility of collusion between the two firms.
- Collusion means that the firms explicitly cooperate in choosing a market price and the division of output between them.
- If the duopolists collude and divide up the market privately, they can produce the monopoly quantity and divide the monopoly economic profits.
- Since the monopoly economic profits are more than the sum of the duopoly profits, the duopolists are better off if they collude.
- When we allow the possibility of collusion the game can turn out differently.

Duopoly Game: Collusion

- In our previous example Firm X and Firm Y can cooperate and agree to charge $14 and to produce 3 units between them.
- They will earn the monopoly profits of $21 in this case.
- There is $1 of additional profit compared to the quantity game and at least $6 of additional profit compared to the price game.
- Any division of this extra profit between the two firms makes both firms willing to collude rather than play the non-cooperative game.
- The possibility of collusion is excluded from the non-cooperative games by the assumption that the firms’ strategies consist of either choosing a quantity or choosing a price.
- Collusion involves choosing a market quantity (or price), production quotas for each member and a division of the monopoly profit between the two firms.

Collusion Problems

- Frequently, side payments are essential to the cooperative solution. Especially when the cartel members have different cost structures.
- OPEC example: Iran and Saudi Arabia.
- Iran’s marginal costs increase more quickly than do Saudi Arabia’s.
- Suppose they do not cooperate and end up at the Cournot-Nash solution: Get profits such that: SA + I = joint
- Suppose they cooperate and implement the monopoly solution: Get profits such that: SA + I = joint
- Since Iran has the crummy marginal cost curve, it will be told not to produce very much in the collusive arrangement.
- Could be that: SA > SA and joint > joint but I > I !
- If joint cartel profit is larger than the joint non-cooperative profit, then there is enough to make side payments to Iran to get Iran’s cooperation.
- Will the side payments be made? Are they legal? Good questions.

Collusion Problems

- Side payments aside, there is also a compelling incentive to cheat on the cartel arrangement.
- Cheating often means that someone is violating the cartel’s production limits - producing more than they agreed to.
- More ends up on the market than was supposed to.
- The price ends up lower than it was supposed to.
- The cartel starts to experience dissention.
- Steps are taken to shore up the cartel agreement.
- This strong internal tendency to cheat led Milton Friedman to once opine that cartels were nothing more than “a flash in the pan.”
- How successful are cartels? How often do they form? Are they able to substantially raise the market? For how long?
- Good questions.

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