Models of Oligopoly (With Graphs and Criticisms)

Models of oligopoly – 1. Cournot’s Duopoly Model 2. Bertrand’s Duopoly Model 3. Edgeworth Duopoly Model 4. Chamberlin’s Oligopoly Model.

1. Cournot’s Duopoly Model:

In 1838, A French economist, Augustin Cournot has developed a model on oligopoly. Cournot’s model dealt with the case of duopoly.

The duopoly model of Cournot is based on the following assumptions:

(1) There are only two producers each owing identical mineral springs. Their waters are identical. This assumption means that the products are homogeneous.

(2) The cost of production of both the firms is zero. The owners operate mineral springs and sell water without incurring any cost of production. The Cournot’s model can be presented when cost of production is positive.

(3) Each firm acts on the basis of the assumption that in spite of his actions and their impact upon market price of the product, the rival firms will continue with the same level of output. In other words, the rival firms will go on producing the same amount of output which they are presently producing.

(4) The market demand for the product is assumed to be linear and the market demand curve facing the two producers is a straight line. Both the firms know the market demand properly.

In Cournot’s model the mineral spring owner will not take into account reactions of his rival in response to his variation in output and thus decides its level of output independently.

Let us suppose that the demand curve of the two producers of the mineral water is represented by the straight line RQ as shown in Figure 9.4. Further suppose that OD = DQ is the maximum daily output of each mineral spring. Thus, the total output of both the springs is OQ = OD + DQ.

It is important to note that the 9.6 total output OQ of both the springs is available for sale in the market at zero prices. It may be further noted that if perfect competition would have prevailed, the long-run equilibrium price would have been zero and actual output produced equal to OQ. Since the cost of production is assumed to be zero the price must also be zero so as to provide a zero profit long-run equilibrium under perfect competition.

At the outset of the model, there is one producer (firm) called A of the mineral water to start the business. Therefore to begin with the only producer will behave like a monopolist. He /She will produce daily OD level of output because the profits will be maximum at output OD and will be equal to ONMP. As it is assumed that costs of production are zero, the whole of total revenue ONMP would accrue as profit.

OP is the price charged by the single producer A. At this stage, let us suppose that the owner of the other spring owner say B enters into the business and starts operating his/her spring. This new producer B sees that the former producer A is producing ON amount of output.

It has been assumed that each firm believes that its competitors will not react to the decision taken by an individual firm. Accordingly the producer B believes that the former producer A will continue producing OD (= 1/2 OQ) amount of output, irrespective of the output he himself produces. In this situation, the new producer B will consider segment MQ as his/ her demand curve lacing him/her. With the demand curve MQ, the producer B will have MR₂, as marginal. The producer B will therefore produce NI (= 1/2 DQ) amount of output. At this point of time, the total output will now be OD + DH = OH, and as a result the price will fall to OP₁.

As the Price level falls to OP1, the total profits made by the two producers will be OIBP₁ which are lesser than ODMP. When OIBP₁ is the total profit, producer A will earn profits equal to ODGP₁ and producer B will earn profit equal to DIBA. It can therefore be observed that the entry of producer B into the market producing output DI by him, the profit of producer A has been reduced.

In this situation, the producer A will have to review the situation. But as embodied in the assumptions of the model, producer A will assume that producer B will continue to produce output DI. Having assumed that the producer B will be producing output DI, the producer A will produce 1/2 (OQ -DI). Thus, producer A will reduce his output.

At this point, the producer B will be surprised as the producer A has reduced the output and as a result producer B will also find that his/her share of total profits is less than that of producer A. Producer B will have to reconsider the situation. But as obvious from the assumptions, producer B will learn nothing from the earlier experience and will again assume that producer A shall continue producing the new current level of output. The producer B will find that he will now be making maximum profits by producing output equal to 1/2 (OQ – New output of A).

The producer B will increase his/her output accordingly. As a consequence of the move of producer B, producer A will find his/her profits reduced. The producer A will again therefore need to reconsider his position and will start producing output equal to 1/2 (OQ – Current output of producer B) assuming that producer B will not react to the change made by producer A.

It is interesting to note that the process of adjustment and readjustment among the producers will continue and as a result the producer A is forced gradually to reduce his/her output and producer B will be able to increase his/her output gradually until the total output OG is produced . The total output produced (OG) is 2/3 of the OQ. i.e.., (OG = 2/3 OD) and each of the producer produces the same amount of output which is equal to 1 /3 OQ.

As represented in the figure, in the final position among the both producers, producer A produces OH amount of the output and producer B produces HG amount of output. In the entire course of adjustments each of the producer assumes that the other competitor will keep his/her output constant at the current level and finds the maximum profit by producing output which is equal to 1/2 (OQ- the current output of the rival producer).

In Cournot’s duopoly model both the producers combine to produce two-thirds of the maximum possible output. Cournot’s model can be extended to those cases of oligopoly where there are more than two sellers. If there were n producers, then according to Cournot’s duopoly solution the total output produced by the n producers would be n/n+1 of the maximum possible output.

Here we would pause to compare the case of Cournot’s duopoly equilibrium with the monopoly and the perfectly competitive equilibriums.

Comparison of Cournot’s Duopoly Solution with Monopoly:

Let us imagine that if both the producers A and B considered in the above model had combined and formed a coalition, then the output produced by A and B together would have be the monopoly output OD. The price in that situation would have be the monopoly price OP. It can be observed that the monopoly output OD produced by A and B together is much less than the output OG produced in Cournot’s equilibrium solution. Moreover, the monopoly price OP charged in case of coalition is much greater than the price OP, determined in Cournot’s model. If we consider that producers A and B earn the monopoly profits ODMP. This profit is greater than the OGFP₂ made by the producers A and B in Cournot’s duopoly equilibrium.

Comparison of Cournot’s Duopoly Solution with Perfect Competition:

Let us again imagine a situation when the market is a perfectly competitive one. In that situation the output would have been OQ and price would have been zero. The price would be zero owing to the assumption that in Cournot’s model the marginal cost is zero.

Accordingly, equilibrium in perfect competition is possible only when the price is equal to zero. In the situation of perfect competition the output OQ is greater than the output produced in Cournot’s Duopoly solution and the price is lower than the price in Cournot’s duopoly solution.

Criticisms:

Even though Cournot’s duopoly model in one of the pioneering efforts made by different economists, the model has been subject to criticism on the following grounds:

(1) Each firm in Cournot’s model acts on the basis of the assumption that in spite of his/her actions and their impact upon market price of the product, the rival firms will continue with the same level of output. This assumption is severely criticized by the critics.

(2) Critics vehemently criticize Cournot’s model for ignoring the mutual interdependence between the duopolistic firms which is the chief characteristic of oligopoly

(3) The assumption of zero cost of production reduces the pragmatic nature of Cournot’s model.

In spite of the criticisms, the Cournot’s model occupies a important niche in economic theory as well as business economics.

Cournot’s Duopoly Equilibrium Explained with the Aid of Reaction Curves:

The Cournot’s duopoly model can be represented with the help of reaction curves. The reaction curves refer to the output reaction curves or price reaction curves. The choice of the reaction curve to be used depends on the adjusting variable used in the model. If output is the adjusting variable then output reaction curve has to be used and price reaction curve has to be used when price is the adjusting variable. In Cournot’s duopoly model, output is the adjusting variable and therefore output reaction curves are relevant.

One has to understand that the reaction curves don’t refer to the reactions which a producer expects will be forthcoming from his rivals but to the producers’ own reactions to the moves of his rival.

2. Bertrand’s Duopoly Model:

In 1883, a French mathematician, Joseph Bertrand formulated his duopoly model. Bertrand criticized Cournot’s duopoly model. Bertrand opined that there was no limit to the fell in price since each producer can always lower the price by underbidding the other and increasing his supply of output until the price becomes equal to his unit cost of production.

Bertrand’s model has some significant differences in the assumptions as compared to Cournot’s duopoly model. The adjusting variable in Bertrand’s model is price and not output. Bertrand assumes that each seller (firm) believes that the rival will keep his/her price constant, regardless of his own decision about pricing. Therefore, in Bertrand’s model both the firms are deemed to face the same market demand and both the firms will strive to maximize own profit assuming that the price of the rival will remain constant.

We will try to understand the Bertrand’s model using the reactions curves of the duopolists. The reaction curves are derived from the isoprofit maps. These isoprofit maps are convex to the axes.

Talking about the isoprofit curve, an isoprofit curve for a specific level of profit is represented on the basis of the different combinations of prices charged by the rival firm. To represent the Bertrand’s duopoly model with the help of the reaction curves, let us suppose that there are two sellers (firms) namely firm A and firm B. The figures 9.5a and 9.5b depict the reaction curves of firm A and firm B respectively. As stated earlier, the reaction curve of a firm is drawn through its isoprofit curve. The isoprofit curve of a firm is convex to the respective price axis. In the figures it can be observed that the isoprofit curves of firm A and firm B are convex to the price axes Pa and Pb respectively.

In figure 9.5a we can see that the firm A can earn a given specified profit from the various combinations of its rival’s and own price. The price combinations at point r, point s and point t on isoprofit curve n π_A1 yield the same level of profit. Now, if firm B fixes its price P_b1, the firm A has the alternatives P_A1 and to attain same level of profits. But, if firm B reduces its price further, firm A has to decide whether to raise its price or reduce its price. Firm A would raise its price above P_A1 when it is at point r and firm would reduce its price when it is at point t where it charges price P_A2 .This adjustment of prices has a limit.

This limit is depicted by point s. Therefore, there is unique price for a firm to maximize its profit. The lowest point of the isoprofit curve represents the unique price. When we join the points r, s, t, we get the curve π_A1. Similarly we have the other isoprofit curves π_A2, π_A3. The reaction curve of firm A can be traced by joining the lowest points of the isoprofit curves π _A1, π _A2, π _A3

The reaction curve of firm B can similarly traced by joining the lowest points of its isoprofit curves as depicted in figure 9.5b.

Bertrand’s model leads to a stable equilibrium. The equilibrium of the duopoly model presented by Bertrand is arrived by incorporating the reaction curves of both the firms A and B in the same figure 9.5c. Point e is the equilibrium as the reaction curves of firm A and firm B intersect at point e. This is a stable equilibrium as because any deviation from this equilibrium point sets in motion forces which will ultimately set the equilibrium at point e again.

Bertrand’s duopoly model has been criticized for the fallacious assumptions. The assumption of the duopolist not taking the reaction of the rival into consideration while deciding its price level is opposed.

3. Edgeworth Duopoly Model:

In 1897 F.Y. Edgeworth, a French economist formulated a duopoly model. Edgeworth criticized the Cuornot’s duopoly solution. He negated the assumption made by Cournot in which each duopolist firm assumes that the rival firm would continue to produce the same output regardless of the level of output produced by the firm.

While formulating the model, Edgeworth has made assumption similar to the assumption made by Bertrand in which the firm assumes that his rival firm will continue to charge the same price regardless of the price set by the firm.

Further one should note that Edgeworth has used the example of Cournot’s mineral springs. Therefore, in Edgeworth’s model the cost of production is zero. One important aspect of Edgeworth’s model is that there is no determinate equilibrium achieved in the model.

Before we move ahead it is important to note that there is major difference between the Edgeworth’s model and the Bertrand’s model. In Edgeworth’s model, the productive capacity of each duopolist firm is limited and neither of the duopolists is able to produce the entire demand. Unlike Edgeworth’s model in Bertrand’s model productive capacity of each duopolist firm is practically unlimited so that the firm can satisfy any amount of demand.

Edgeworth’s model of duopoly has been represented in Figure 9.6.There is an assumption in the model that the products of two duopolists firms are completely identical. As a implication of the assumption the market would be equally divided between the two duopolists at the same price of the product.

Let us suppose that there are two firms 1 and 2 in the market. The size of the entire market is B₁ B. The entire market B₁B is divided into equal parts between the two firms. Both the firms face identical demand curves. Firm 1 has the demand curve QX and Firm 2 has the demand curve QXb. It can be observed from the figure the firm 1 has the maximum capacity of OB amount of output and firm 2 has the maximum capacity of output OB₁. Price is measured and represented along OQ ordinate.

At the outset let us imagine that there is a single firm, the firm1 in the market. Being the only firm the firm will behave as a monopoly. The firm will therefore sell OA amount of output and charge a price equal to OP. The firm earns profit equal to OPMA. This is owing to the assumption that the cost of production is zero. Let us suppose that at this point the other firm viz., the firm 2 enters the market and assumes that the firm1 will not change its price as the firm1 is enjoying maximum profit and thereby firm2 sets its price slightly below firm1’s price (OP). By doing this firm 2 is able to sell its total output and also capture a substantial part of firm 1’s market.

Now as the firm 1 realizes the reduction in its sale, it will set its price slightly below the price of firm2 in order to regain the lost market. This will ensue a price war between the firms. The price war takes the form of price cutting which continues until the price reaches OP₁. At the price level OP₁, both the firms are able to sell their entire outputs. The firm1 will sell OB and firm2 will sell BB₁ at this price level. The price level OP₁ seems to provide stability to the model. But Edgeworth in his model explain that this price level is not stable.

There is an inherent instability in the model. The cause for the instability in the model is that, each firm realizes that its rival is selling its entire output and it will therefore not be able to alter its price and each firm thinks that it can raise his price to OP and can make profit. This again sets the chain of actions and reactions. If the firm 1 raises its price to OP. Firm 2 will assume that firm 1 will retain its price level at OP and finds that if it raises its price to a level slightly below OP it can sell its entire output at a higher price and make profit.

As a reaction to the move of firm 2, the firm 1 will reconsider his situation and react. Firm 1 observes that its price is higher than the price of firm 2 and its sale has reduced. Again assuming firm 2 to retain its price, firm 1 will reduce its price slightly below the price of firm 2. The price war between both the firms resumes. This price war continues and as a result the price remains unstable and keeps on fluctuating i.e., moving up and down between OP and OP₁ where OP is the monopoly price and price OP₁ is the competitive price. There is no unique equilibrium in Edgeworth’s duopoly model.

We can thus conclude that in Edgeworth’s duopoly model the equilibrium is indeterminate. The price in Edgeworth’s duopoly model constantly oscillates between the monopoly price and competitive price. The assumptions of the Edgeworth’s duopoly model have been criticized by the critics.

4. Chamberlin’s Oligopoly Model:

As compared to the classical Oligopoly models of Cournot, Bertrand, and Edgeworth the Chamberlin’s oligopoly model is comparatively more advanced and superior. Chamberlin’s model is based on the assumption that the oligopolistic firms understand and recognize the mutual interdependence and behave accordingly.

Chamberlin opined that the oligopolists are intelligent enough to recognize their interdependence and they therefore jointly produce monopoly output and charge monopoly price. Thus, in Chamberlin’s model the stable equilibrium and maximization of joint profit by the oligopolist is accomplished. It is interesting to note that the oligopoly firms behave in a non-collusive manner.

We have tried to illustrate the Chamberlin’s oligopoly model in figure 9.7 Chamberlin assumes that there are two producers viz., producer 1 and producer 2. The cost of production has been assumed to be zero and the product produced is homogeneous. Further, the market demand curve DD₁ has been assumed to be linear.

In order to understand the Chamberlin’s model we assume that producer 1 enters the market and is the first to start production. Producer 1 faces the linear demand curve DD₁ representing the whole market. MR₁ is the corresponding marginal revenue curve. Producer 1 will produce OD₂ which is half of OD₁ which is equal to the monopoly output and fix monopoly price OP.

Let us suppose that at this point of time the producer 2 enters the market. The producer 2 assumes that producer1 would continue to produce OD₂ output and thus MD₁ portion of market demand curve is the relevant demand y curve for producer 2 and the corresponding marginal revenue curve for producer 2 is MR₂. Producer2 will find it profitable to produce half of D₂D₁ which is equal to D₂R. This is owing to the assumption that marginal cost of production is zero.

Now the total output available in the market becomes equal to OR. The total output OR includes OD₂ produced by producer1 and D₂L produced by producer 2. When the total output becomes OR the price will fall to OP₁. As a result, producer 2 would earn profit equal to D₂RNQ. The profit of producer 1 would fall from OD₂MP to OD₂QP₁.

Up till now the producers in the Edgeworth’s model have been behaving in the similar way as they behaved in Cournot’s model. At this juncture and onwards the Chamberlin model would deviate.

The producers in the Chamberlin’s model learn and realize from the earlier experience about their interdependence.

As a result of this awareness of mutual interdependence, producer 1 would decide to produce output OS equal to output D₂L of producer 2. The total output of both the producers is equal to monopoly output OD₂

When the aggregate output becomes OD₂ the price level would rise to QP. In Chamberlin’s model the Producer 2 is also aware of the mutual interdependency. Producer 2 also understands that the interests of both the producers are best served when they produce half of monopoly output individually and hence would retain its output at D₂L.

Therefore, it can be observed that in Chamberlin’s model the duopolists realize their mutual interdependence and behave intelligently. A stable equilibrium is ascertained in Chamberlin’s model wherein the duopolists combine to produce monopoly output and charge monopoly price.

Chamberlin’s duopoly model is subjected to criticism even though it is an improvement over the other classical models on Oligopoly. Critics fail to appreciate the maximization of joint profits without collusion in Chamberlin’s model. Even in formal collusion there is an inherent tendency of collusion partners to undercut each other’s prices.