The Stackelberg equilibrium is when the leader, firm one, chooses a point on firm two’s reaction curve that touches the firm one’s lowest possible isoprofit line, thus yielding the highest possible profits for firm one.

Cournot

A problem that arises with the Stackelberg model is that it assumes that one firm is able to make its decision before the other firm. The Cournot model is a static one and is used when two firms are simultaneously trying to decide what quantity to produce. In this scenario, firms forecast the other firm’s output in order to make a sensible output forecast for their firm. The Cournot-Nash model is based on mutually consistent expectations. A Nash-equilibrium can be interpreted as a pair of expectations about each firm such that, when the other firm’s decision is revealed, neither firm wants to change his behaviour.

The total output of the industry would yield a market price of where e is the expected output. This leads to the profit maximisation problem () and functional relationship (firm one’s reaction curve: and firm two’s reaction curve: ) similar to the Stackelberg model. The Cournot equilibrium combination of output choices satisfy and as is the optimal output level for firm one, assuming that firm two is producing , and the optimal output level for firm two is , assuming that firm one keeps consistent and produces at .

The Nash solution will always emerge in this situation because if the two firms decided to produce at a level of outputas illustrated, firm one will cut its production to settle on its own reaction function and the new outputs will be. In response to this, firm two will increase its production such that the new pair of outputs will be. A similar process will repeat itself until the stable equilibrium level of is reached.

Additionally, if there were several firms involved in Cournot equilibrium, an n-firm concentration ratio, each firm would have to take into account the expected output choices of all the incumbent firms and the total output of n firms would therefore be. The profit maximising condition where marginal revenue equals marginal cost would lead to:

where si is firm i’s share of the total output market. From this it can be seen that a monopoly is a specific case of an oligopoly as the monopolist would have the entire market share and tends to zero.

Bertrand

The Bertrand model is one of simultaneous price setting and is related to Cournot’s model, but the strategic variable is price rather than quantity produced. The Nash equilibrium concept is similar as the solution of the game is to find a pair of prices such that each price is the profit maximising choice for each firm and such that these choices are all consistent. Taking the standing assumption so far in this essay, firms selling homogenous products in the Bertrand model will find an equilibrium point where price equals marginal cost, which is the same point reached in a perfectly competitive industry.

The line of reasoning for the Bertrand model is based on two points, firstly, all firms set the same price in equilibrium and secondly, price equals marginal cost. If two firms therefore were selling a truly homogenous product, a price increase from one firm would mean that the other took the entire market share. Alternatively, it priced the good slightly below its rival then that firm will capture the entire market share. Thus it must be true that two firms in equilibrium must be charging the same price (p*) for the homogenous product. Additionally, if the equilibrium price was greater than the marginal cost, one firm can capture the whole market share and still make a profit if it charges p*-e where e is a fraction of the price. The other firm would respond by cutting its price to rival that of the other firm up until both firms price the product at marginal cost of production. It is often see that competitive bidding among firms that are not able to collude can result in prices that are significantly lower than process seen in other theory of the firm models.

Examples and Implications

A typical example of the Bertrand model is that of shops and bars that publish non-negotiable prices. This is not very popular as firms would still prefer to make a significant profit rather than undercutting the other incumbents on price. A popular example of the Stackelberg model is Google who is a significant large player in the internet market. It started off providing a search engine but now has diversified into maps and emails which Microsoft have tried to imitate. Lastly, Cournot competition is widely seen in many oligopoly markets and a prime example is the aircraft industry where there are initially high set up costs and only two providers, Airbus and Boeing.

A comparison of the three models shows that the Stackelberg price is greater than the Bertrand price but lower than the Cournot. Stackelberg aggregate output and consumer surplus is greater than Cournot, but it is lower than Bertrand aggregate output and consumer surplus. Stackelberg aggregate output is also greater than a pure monopoly or cartel but less than a perfectly competitive output. Finally, Stackelberg price is lower than a pure monopoly or cartel but greater than the perfectly competitive price.