Coursework

A critique of Cournot, Bertrand and Stackelberg models of oligopoly

The Cournot, Bertrand and Stackelberg models, commonly known as the conjectural variation models, intend to explore the behaviour of firms in an oligopolistic market structure.

During the course of this essay I shall outline the basic framework of each model and present an overview of their most significant limitations. In addition I shall compare and contrast the models and provide examples of how economists have subsequently developed them. None of the models realistically explains firm behaviour over time, nor do they consider collusive theory. In this sense it can be argued that the models are fundamentally flawed. However, the Cournot model has made a significant contribution to both oligopoly theory and noncooperative game theory, whilst the Bertrand and Stackelberg models have successfully overcome various limitations in the original Cournot model. One can also find evidence of the application of conjectural variations models in areas as diverse as environmental policy and welfare economics.

An oligopoly is characterised by very few firms controlling the market, for example the United Kingdom supermarket industry. Each model which is studied in this paper shares the common assumptions that firms operate in an oligopoly, in which there are barriers to entry (factors which prevent other firms entering the market) and no exit of existing firms. There exists strategic behaviour, in which one firm will be concerned about the behaviour of other firms in the market. A key assumption is that of noncooperation; firms will not collude to maximise joint profits. Consumers are assumed to have complete information i.e. they are aware of all firms and their prices and view products as homogeneous (identical). It is also assumed that they do not face transportation costs. Thus a consumer’s choice of product is solely determined by its price. One should note that with the exception of strategic behaviour, none of these assumptions may hold in a typical oligopolistic market.
The models are distinguished from each other by two factors: the decision variable (the variable which firms are competing on) and/or the timing of firms’ actions. Each model assumes that a firm cannot set both price and quantity. In the Cournot and Stackelberg models, the decision variable is quantity, whereas the Bertrand model presents price as the decision variable. With regard to timing, we could treat each model as a one-period game, in which the firms are the ‘players’ and their ‘moves’ are either setting price or quantity, as dictated by the specifications of the model Therefore, in the Cournot and Bertrand models, the firms move simultaneously whereas in the Stackelberg model there is one leader and other firms are presented as followers.

The Cournot Model

The Cournot (1838) model is the earliest known attempt at describing oligopolistic behaviour. Since firms in this model are competing on quantities, each firm charges the market price. This is assumed to be determined from the inverse demand function (or perhaps an auctioneer) as the price that equates the amount demanded by producers to the amount produced by the industry.
The inverse demand function gives price p charged by each firm as a function of total industry output Q Each firm also faces a cost function C(qi), where qi is the quantity chosen by firm i. Thus the profits of each firm, given by Πi, can be calculated as total revenue pqi minus total cost C(qi)qi. Each firm attempts to maximise profits. An equilibrium consists of a set of quantity choices by each firm such that no firm can increase profits, given the quantities chosen by other firms.
A simple version of the Cournot model is presented below. Here, there are two firms – firm i and firm j. Each firm faces a linear demand function and a cost function which equals constant marginal cost c

p = a – Q
Q = qi + qj

Since marginal cost equals c, it follows that Πi = pqi - cqi

We assume that dqj/dqi = 0 and that profits are maximised where dΠi /dqi = (a-c) - 2qi - qj = 0.

Therefore firm i will set quantity qi = ((a-c) - qj)/2

This is known as firm i’s ‘best-response function’. It expresses the profit-maximising quantities of the firm as a function of quantities chosen by its rival. Since firm j faces the same costs, its best-response function, by symmetry, is qj = ((a-c) – qi)/2. By substituting the best-response function of firm j into that of firm i, and vice- versa, we can find quantities which are best responses to each other, and therefore reach an equilibrium. In this case, a unique equilibrium is characterised by each firm setting their quantities equal to (a - c)/ 3.

The monopoly quantity for firm i (where the firm captures the industry) is found by setting qj equal to zero in the firm i best-response function. Thus the monopoly quantity for each firm can be ascertained, and is given by (a –c)/2. Note that if each firm cooperated, they would produce (a –c)/4 and therefore share the industry However, this outcome is not considered an equilibrium of the game. In the ‘competitive’ outcome, firms would set price equal to marginal cost. Thus p would equal c, resulting in Q = a – c. Each firm would produce half of the competitive output level, equivalent to (a – c)/2 This outcome is socially optimal, since the value placed on the last unit sold by the producer is the same as that placed on it by the consumer.

The model can be extended to become a multi-period model, in which each firm’s quantity choice eventually converges to the Cournot equilibrium. It can also be extended to more than two firms, which predictably results in a convergence to the competitive equilibrium, with falling profits and prices for each firm and increasing industry output.

The outcome of this simple duopolistic Cournot model signifies that in equilibrium, firms will set price higher than marginal cost and therefore make positive profits. Due to its representation as a one-period game, the Cournot outcome assumes that firm j will not change its behaviour in response to a change in the behaviour of firm i, signified by the fact that dqj/dqi = 0. This conjecture is patently false as firms would in reality observe across time that they can influence each other through changes in their behaviour and subsequently alter their quantity choices accordingly.

The Bertrand Model

The Bertrand model of duopoly is assumed to have the same characteristics as the Cournot model, with the exception that firms compete in price whilst allowing industry demand to determine quantity. Again firms are assumed to act simultaneously upon interaction with one another.

In the simple model presented below, each firm faces a constant marginal cost c. Note that if firms were to set price lower than marginal cost they would face losses. Also, there exists a profit maximising monopoly price which is the maximum price that firms could be expected to charge. A firm could potentially capture the industry by setting price one unit lower than its rival, or share the industry by setting price equal to its rival’s price.

This gives rise to the following set of best responses for player i:

1.If firm j sets price less than or equal to marginal cost, firm i will set price equal to marginal cost and neither firm will make profits.
2.If firm j sets price greater than marginal cost but less than the monopoly price, firm i will set price at pi = pj - x, where x is greater than 0.
3.If firm j sets price greater than or equal to the monopoly price, firm i will set price equal to the monopoly price.

There is a unique equilibrium where each firm’s price maximises their profit given the price set by the other firm. This is where pi = pj = c. To see this note that if one firm sets a price lower than c it will make losses and if it sets price higher than c then the other firm will capture the market. In stark contrast to Cournot, this model states that in equilibrium firms set price at the socially optimal level, and do not make profits.

The Bertrand model, unlike the Cournot model, provides a formal mechanism for determining prices. However, the outcome improbably signifies that a firm could capture the market by simply setting price one unit lower than its rival. Hotelling (1929) developed a model in which products were identical in price but differentiated by location and subsequently reached the more realistic conclusion that one firm will not necessarily capture the market by pricing one unit of currency lower than the other firm. As with the Cournot model, the static framework of the Bertrand model leads to the unrealistic assumption that rivals will not play a repeated game in which they change their behaviour in response to changes in the behaviour of rivals.

Bertrand and Cournot Models compared

If firms find that price is easier to adjust than quantities, they are more likely to accept the price determined by the market. Therefore one would expect them to set quantities rather than price, resulting in a Cournot type situation. This is arguably the case in reality, with the exception of industries such as insurance, in which prices are more difficult to adjust than quantities.

The Stackelberg Model

The Stackelberg (1934) model, like the Cournot model is based around best-response functions and shares the same characteristics. However, the original Cournot model is set up as a ‘simultaneous move game’. In reality, firms may not necessarily set quantities at the same point in time and the Stackelberg model overcomes this by setting the game up as one of sequential moves, in which there is a ‘leader’ and at least one ‘follower’. The leader assumes that its rivals will set quantity according to their best-response functions and can subsequently maximise its own profit by choosing its best response to the rivals’ best response functions.

A simple version of the Stackelberg model, which enables direct comparison with the Cournot duopoly model described previously, is presented below:

Recall that the best-response function of firm j is ((a –c) – qi)/2.

If firm i is the leader, we assume that it can choose its best response to the best-response function of firm j.

Therefore Πi = pqi - cqi = (a – (qi + ((a –c) – qi)/2))qi - cqi.

Profits are then maximised where Πi = 0 i.e. where qi = (a – c)/2, which is the monopoly quantity.

Substituting this value back into the best-response function of firm j, we find that qj = (a – c)/4. Thus the leader produces more than the follower and obtains higher profits. The Stackelberg model therefore implies that there is a ‘first mover advantage’.

Although the behaviour of firms is presented as sequential actions which take place over time, it is still a static model in the sense that the variable time does not directly appear in the model. Negishi and Okuguchi (1971) developed a Stackelberg model in which time is made more explicit.

The Stackelberg model, like the Cournot and Bertrand models, does not hold up to close scrutiny when extended to multiple time periods. Each firm in the duopolistic Stackelberg model is expected to alternate between leader and follower ad infitum. The model does not take into account the fact that over time, each firm will build a memory of its past actions and will therefore attempt to gain a first mover advantage in every period, thus there is no justification for such a symmetric pattern to emerge. The concept of ‘memory’ or history-dependence has been developed by Chamberlin in his analysis of repeated games, or ‘supergames’ as they are commonly known.