Monopoly and Duopoly Models

In class, we played a monopoly game. There were 14 students, and 14 cards were distributed to assign roles. A King, Queen and a 2, 4, 6, 8 and 10 (one set of hearts, one of spades) was shuffled and distributed. The King became the recorder, the Queen the monopolist. The numbered cards represented redemption values. Each student individually negotiated a price with the relevant monopolist. Two strategies were evident. One monopolist reduced the price as the round proceeded and negotiated a deal with every buyer. The other monopolist chose not to strike a deal with two potential consumer and instead kept the price uniformly higher than the other monopolist. The second strategy generated the higher profit. If we had repeated the game (randomize on redemption values and renegotiate) then the strategy of lower prices as the round proceeded would break down. Students would learn to wait until the end of the round to negotiate and all would claim to have low redemption values. This is the process presumed by traditional theory. The monopolist sets one price because he or she can not predict the reservation values of consumers and can not successfully negotiate individually. Instead, a single profit maximizing price is set. With linear demand and constant marginal cost, the theory is simple, the profit maximizing rectangle connects the midpoints of the triangle formed by demand and marginal cost. (See diagram,) The height of the rectangle is profit per unit and the width is the number of units. The product is profit and is also area. (What would a monopolist do that can predict each consumer’s reservation value? )

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Cournot Games

In this game, you and your partner simultaneously choose outputs and record your profits below. So, cover your output choice with your hand and then reveal the choice. For example, if they choose 2 and you choose 6, then your profit is 24. You are welcome to talk to each other. There are a number of ways to play the game. Cournot assumed people and firms would play as if their choice had no impact on the other player. This makes the game simple. For example if your partner choses 3 then your “best response” is 5 because your profits would be 20, any other choice reduces your profit. (We break ties by choosing the higher output.) Furthermore, since you aren’t trying to change the other player’s behavior, there is no other consideration. Alternatively, you can try to manipulate the choice of your partner. Obviously it is best for you if they pick 0 and you pick 6. But watch out, if you try to manipulate them, then they may try to manipulate you. The table is constructed from a simple demand function, p = 12 - (Q1+Q2) where Q1 and Q2 are the output choices of you and your partner. The profits reported are just pQ1, that is, I assumed zero cost. For example if your partner chooses 3 and you choose 5 then the output is 8, the price is 12- 8 = 4 and your profit is 4x5 = 20. A bit more formally, firm one’s profit is Q1(12-(Q1+Q2)).
My Firm’s Profit Table
| | |Other Firm’s Choice |
| | |0 |1 |
|1 | | | |
|2 | | | |
|3 | | | |
|4 | | | |
|5 | | | |
|6 | | | |
|7 | | | |
|8 | | | |
|9 | | | |
|10 | | | |

The table above is just a linear demand curve in table format. Let my firm choose to produce one unit. Then if the other firm produces zero units, the price is 12-1 = 11 and revenue is 11*1 = 11. Every time the output of the other firm goes up one unit, the price falls one unit and my revenue declines one unit. Therefore the sequence is 11, 10, 9 … If I produce two units and the other firm produces zero units then the price is 12-2 = 10 and revenue is 10*2 = 20. Every time the output of the other firm rises by one unit, price declines by one unit and revenue falls by 2, so the series is 20, 18, 16… And so on.

Monopoly: If the other firm produces zero, the profit max is to produce half of 12, or 6. The price is 6 and the revenue is 6*6 = 36.

Shared Monopoly: This can be shared equally if both choose to produce 3 units. Then the price is still 6 but revenue is 18 for each firm. The difficulty is that each firm has an incentive to cheat. The best response to 3 is not 3 but 4.5.

Cournot-Nash Equilibrium: The only point were both firms play their best response to the other’s choice is 4,4 – this is the Nash equilibrium discovered by Cournot roughly two centuries earlier. Notice that the Cournot equilibrium and the shared monopoly form a prisoner’s dilemma.

Stackelberg leader: if the other player can be relied on to always play their best response, then the leader will pick 6. The follower picks 3. The leader’s profit is 18, the follower’s is only 9. The leader gets the shared monopoly profit with the added advantage that the other firm gets only half that.

Stackelberg warfare: If the follower gets fed up and decides to also play 6, both get zero. This is not strictly rational (unless the follower believes the leader will eventually change strategy) but is very human.

As we saw from the prisoner’s dilemma game, behavior is more complicated than theory suggests and is difficult to interpret. Some cooperate and form a shared monopoly. Some see the opportunity to push followers into a corner. Some seem to play a mixed strategy. Issues of fairness may be important in this game as well as in the prisoner’s dilemma game.

We will discuss the issue of monopolies in essential goods. This is the topic of current experiments and will not be written up.

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