Classical Models of Oligopoly

1. Cournot’s Duopoly Model

Cournot founded the theory of duopoly. His duopoly model consists of two firms marketing a homogenous good. Cournot uses the example of mineral spring water, whose production costs nothing. This is convenient, but not necessary. His model can extend to accommodate production costs and so, we will temporarily assume that production costs rise with the output of each firm.

Algebraically:

C_i = c_i q_i²                             …[1]

where C_i is the total cost of the ith firm, c_i = is a parameter and ∂C_i/∂q_i = marginal cost = 2c_iq_i

Each firm is aware of the market demand curve. It consequently knows that the market price will be affected by its sales decisions as well as the sales of its rivals. Thus each firm is aware of the demand curve –

p = a – b (q₁ + q₂)            …[2]

where a, b > 0.

Each firm is a profit maximizer, choosing a level of output which will maximize its profit. We know that its profits are maximized where its marginal cost equals its marginal revenue, so that its equilibrium output will be characterized by MR_i = MC_i.

Properties of Cournot Equilibrium

In Cournot Equilibrium, the industry’s output is less than that under perfect competition. Correspondingly, the Cournot price is higher than in perfect competition. On the other hand, the Cournot equilibrium industry output is more than under simple monopoly, and the Cournot equilibrium price is lower. Thus the Cournot solution of the oligopoly action-reaction problem lies between the situation in perfect competition and monopoly.

Limitations of Cournot’s Duopoly Model

The limitations of Cournot’s model provided a launching pad to many later models of duopoly.

Cournot’s duopolist can be characterized as naive since he never learns from his experience which shows his rival’s sensitivity to his own sales decisions. Stackleberg develops a model designed to overcome this naivete.

Cournot’s oligopoly’ s smooth continuum stretching from monopoly at one end to perfect competition at the other has also come under question. This continuum implies that the difference between the four market structures is a difference of degree and not of kind. Chamberlin finds this unconvincing. Hence he offers an alternative model where behavioural changes separate the Cournot oligopoly from monopoly and perfect competition.

One of the earliest criticisms of Cournot’s model has been that it fails to allow for the incentive to a firm to lower its price and enlarge its sales. Bertrand suggested that instead of the price equilibrium envisaged by Cournot, a duopoly would experience competitive price undercutting. This would better explain the occasional price wars that flare up in markets with a few sellers.

2. Bertrand’s Duopoly Model

Cournot assumes that the duopolist takes his rivals’ sales as constant while making his decisions. Bertrand suggests that it is more plausible to assume that the duopolist takes his rival’s price as constant.

Price War

If the duopoly firm adopts Bertrand’s conjecture, it thinks that by marginally cutting its price (with other prices given) it can capture the entire market. An expansion in its sales is attractive to the duopolist so long as price exceeds marginal cost, since every extra unit sold add to its operating profits. The other firm too responds in kind and a price war flares up. The process of competitive price cutting to capture the market goes on as long as the price exceeds the marginal cost of the firm at its current sales.

This is illustrated in Fig. 1. The initial price p exceeds the marginal cost of firm A and firm B. This triggers off competitive price cutting and the price is continually pulled down. For instance, at p₁ the output at which MC_A= MC_B = p₁, is greater than market demand and sales. Hence sales will be less than desired. At less than the desired level of sales, the marginal costs of A or B or both are likely to be less than p₁. This gives them the incentive to expand sales by further price cutting. Thus price war continues until the competitive price p_c is reached.

Competitive Equilibrium:

At the competitive price, the market demand equals the combined supply of the two firms, and their marginal costs equal the market price. In the Fig. 1, the competitive price is represented by p_c. At this price, A supplies a of the market and B supplies b.

However, equilibrium at the competitive price is not guaranteed in Bertrand’s model. If the price falls below p_c, it may remain there inspite of excess demand. This is because whoever raises the price in response to excess demand will forfeit his share in the market.

The Edgeworth Variant

Edgeworth suggests a variant wherein neither firm has sufficient capacity to supply the whole market at the competitive price. In such a situation, a firm which raises its price will still have a part of the market to cater to. Behaving as a monopolist in this part of the market, this firm may charge a higher price. This successful price hike will encourage the other duopolist to follow suit.

Once the price exceeds their marginal costs, the temptation to undercut the other’s price will again prove too strong. Thus price wars resume, returning the two sellers to the competitive price. This renews the entire process. Thus, in the Edgeworth variant, there is only an endless oscillation between the competitive price and a higher price.

Product Heterogeneity

It has been pointed out that the price war is intense in Bertrand’s model because the produce is homogenous. With a homogenous product, the low priced firm captures the entire market. Only a part of the market demand will swing to the low priced firm if the product is heterogenous. Hence product heterogeneity will reduce the gains from price-cutting. In such situations, price cutting will not lead to the competitive price, but will halt at a higher price.

Limitations of Bertrand’s Duopoly Model

Although Bertrand’s model explains price wars successfully, these are relatively infrequent in industrial markets. Actually as Rothschild says, “Price rigidity is an essential aspect of ‘normal’ oligopostic strategy”.

This does not deny the importance of price wars which are occasional. As Rothschild argues, it is the ‘fear’ of price wars and preparation for them that marks oligopolistic behaviour even in normal times. Hence, what calls for explanation is not the price wars themselves, or their consequences, but how oligopolists try to avoid them or prepare for them. It is these measures, “aggressive or defensive which are peculiar to oligopoly.”

On these measures, the Bertrand model is silent. Bertrand’s model, just like Cournot’s model, assumes naivete on the part of the duopolists.

As Chamberlin says “When a move by one seller evidently forces the other to make a countermove, he is very stupidly refusing to look further than his nose if he proceeds on the assumption that it will not”. Since in both models, the firm assumes that no countermoves will be made inspite of its experience to the contrary, its behaviour is irrational. To overcome this irrationality, Chamberlin suggests an alternative oligopoly model.

3. Chamberlin’s Small Group Model

Adam Smith had once maintained that people in the same line of business attempt to collude whenever they get together. Chamberlin suggests that this would be so, when the sellers are very few.

With very few sellers it becomes possible for an oligopolist to take into account the reactions his decisions evoke in his rivals, and the effects of their reactions on the market price and output. In other words, the single oligopolist can take into account both the direct as well as the indirect effects of his decisions.

Equilibrium Solution

The knowledge of the indirect effects of his action will convince the oligopolist that it is better to charge a monopoly price and share monopoly profits with other sellers than to chase elusive individualistic profits which disappear under the weight of action and reactions of oligopolistic rivals. Thus, each oligopolist, with his knowledge of the market demand curve, calculates the monopoly price, and maintains it.

The monopoly price depends upon the market demand as well as the aggregate marginal costs in the industry. Hence each oligopolist must also be aware of the marginal costs of other firms. Chamberlin seems to have assumed that marginal costs of the different firms are identical. On this assumption, it is a simple matter to calculate the monopoly price in the industry as shown in Fig. 2. Here the point of intersection of the aggregate MC and MR gives the industry’s output. The corresponding price is the monopoly price. The firm’s own output is calculated by equating its marginal cost to the MR in the industry.

Chamberlin, Cournot and Competition:

When the number of sellers is very small, they may consider all the indirect effects of their output decisions, in which case the Chamberlin solution results.

The full knowledge of indirect effects is less likely, the larger the number of sellers in the market. Hence as the group size increases, oligopolists may fall back on the Cournot assumption of zero conjectural variation of output, in which case the Cournot solution obtains.

Finally, when the group size becomes too large, each seller may consider even the direct effect of his action to be negligible and ignore it. This takes the market towards perfect competition. Thus, according to Chamberlin, behavioural shifts accompany changes in the number of sellers, separating the three models.

4. Stackelberg’s Duopoly

Stackelberg introduces sophistication into the Cournot model. From experience, each seller becomes aware that his rival reacts to his sales plan. He then estimates his rival’s reaction curves and introduces them into his own calculations when trying to choose a profit maximising output. Algebraically, the firm’s profit maximising output is given by:

where dq₂/dq₁ is estimated from the reaction curve of the Cournot duopolist.

The firm which allows for its rival’s reactions in its decision making is called a leader by Stackelberg. A firm has also the option of turning a Nelson’s eye to his rival’s reactions and behaving blindly as a Cournot firm would. In that case, it would be called a follower.

Now, a Stackelberg firm would compare its prospective profits in a follower’s role with those obtaining from a leader’s role. It will choose the more profitable role. If both firms choose to be leaders, the model breaks down. If only one chooses to be the leader, the Stackelberg solution results. The Cournot solution obtains when both firms choose to be followers.

5. Sweezy’s Kinked Demand Model

Unlike the other models, Sweezy’s kinked demand model applies to the case of heterogenous products. Since products are heterogenous, every oligopolist faces a downward sloping demand curve for his product. However his demand curve appears to be kinked at the going price to the oligopolist.