# Duopoly

A duopoly is a market form, more a special form of an oligopoly. The assumptions are that there are two firms that sell their product to a large number of customers. They produce a homogeneous good at constant marginal costs without any fixed costs. The question is if there is a Nash equilibrium. That means that the action of one player is an optimal response to the other player's strategy. If the equilibrium is reached, no one of the participants wants to deviate, then they have made the best choice. Both firms want to maximize their profit. You can differentiate between two Duopolys: You can compete in quantity or price: Cournot Duopoly or Bertrand Duopoly .

## Cournot Duopoly

If you compete in quantities then it is called a Cournot duopoly. The price determines by the demand function. Both firms choose the quantity that they want to produce at the same time. The payoff of each firm depends on this decision. The nash equilibrium is permitted through the solution of the reaction functions of both firms. If the firms would pool, then there can be reached a higher level of welfare for the consumers and in total under the assumption that the goods are complements .

### Stackelberg Duopoly

A special form of the Cournot Duopoly is the Stackelberg Duopoly. It is an extension of the Cournot duopoly. That means that one of the two firms has the first-mover advantage. The first mover is called Stackelberg leader and decides about their quantity before the second firm. This one is called Stackelberg follower. The first firm knows that its action is observable by the second firm. For getting a solution by calculation you use a backward induction .

## Bertrand Duopoly

Bertrand invented a new kind of duopoly. The assumption is that firms use the quantity as a strategic variable and therefore adapt the price. If the firms compete in directly in prices the name is Bertrand competition. The Strategy is that both firms choose their price simultaneously. The clients buy where the price is lower.The only possible equilibrium is if both prices (p1 and p2) and the fixed costs are the same. But consequently, it leads to losses while a higher price leads to profits of zero. This is known as the Bertrand paradox. Literature shows ways out of Bertrand's paradox: for example asymmetric marginal costs or switching costs. Also different different preferences of the clients could be a solution . Another analysis of the Bertrand Duopoly with product differation leads to profits. In equilibrium the profit of each firm increases by more detailed information of its own. If you have perfect substitutes goods then pooling leads to a lower level of welfare for the consumers and in total .

## Comparison of the duopolys

If you compare those two kind of duopoly possibilities there are some important issues.

• Bertrand vs. Cournot
1. Cournot profits are strictly positive.
2. Fixed costs have no impact only if the equilibrium profit would be negative.
3. Quantities and profits decrease in the number of competing firms.
4. Bertrand case is strictly better than Cournot regarding the welfare.
5. Social value of information is positive in cournot case and negative in bertrand case.

The overall surplus is higher in Cournot when the goods are substitutes. If the goods are independent, then the profits are equal .

• Stackelberg vs. Cournot

In the Stackelberg case the first mover has an advantage. The leader receives a higher profit than the follower. If the products are substiutes then the pricing will be competitive. If the products are less substitutes then the pricing gets less competitive. Also if the marginal cots get higher in the slope. Additionally, competition arises if the costs have no negative effect on costs when products are quite similar .

### Summary of the types

So there are lots of different possibilities of equilibrium outcomes. The consequences of each duopoly is depending on the setting of the environment. The solution depens on some factors: type of information, conditions of the goods and the degree of product differentiation .

## Exampels in the Economy

One of the most known examples in real life is the competition between the companies Airbus and Boeing. Because of the high amount of orders there's no need to lower the costs. It approaches the Cournot price. The question is if there's not enough competition which impact has it on the clients. The two firms can hold the prices very high. For such possible collaborations, there are authorities to control the behavior of the companies .

## Footnotes

1. Löffler, C. (2006), pg. 41-44
2. Vives X. (1984), pg. 87
3. Blum U. et al. (2006) pg.69
4. Löffler C. (2008), pg. 43-46
5. Vives X. (1984), pg. 79-87
6. Vives X. (1984), pg. 76-93
7. Mukheree A. et al. (2012), pg.555-557
8. Löffler C. (2008), pg. 117-123
9. Vives X., (1984), pg. 93
10. Röhl K. (2018), pg.1-3

 Duopoly — recommended articles Price-Taker — Monopson — Price taker — Demand — Discriminatory pricing — Price sensitivity — Market mechanisms — Free competition — Marginal pricing