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List of oligopoly models: 1. Cournot’s Duopoly Model 2. Bertrand’s Duopoly Model 3. Chamberlin’s Small Group Model 4. Stackelberg’s Duopoly 5. Sweezy’s Kinked Demand Model.
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:
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Ci = ci qi2 …[1]
where Ci is the total cost of the ith firm, ci = is a parameter and ∂Ci/∂qi = marginal cost = 2ciqi
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]
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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 MRi = MCi.
Conjectural Variation:
The firm’s marginal costs can be simply estimated since they depend only upon its own output. But its marginal revenue depends not only on its own output but also on its rival’s output, and how its rival changes its output in reaction to the firm’s decisions. This is shown by the second term on the right hand side in the following –
Thus, in order to estimate its marginal revenue, the duopolist is forced to conjecture about how his rival is likely to vary its output in response to his own decision. That is to say, he conjectures about the sign and size of ∂q2/∂q1.
The Cournot Assumption:
Cournot assumes that this conjectural variation (∂qj/∂qj) is zero. Thus the Cournot firm assumes its rival’s current output to be constant in face of variation in its own supply.
On the basis of its rival’s current output, from the market demand curve, it estimates its own marginal revenue function to be –
Equilibrium:
At the profit maximising output of each Cournot firm, marginal revenue equals marginal costs. Hence from eqns. [3a], [3b] and [1], we get –
It can be seen that because each firm calculates its marginal revenue on the basis of the current output of its rival, its rival’s current output determines its output decision. Of course, its output decision also depends on its own costs as well as the market demand conditions (represented by a, b, ci).
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The eqns. [4] and [5] are called reaction curves. They express the dependence of each firm’s profit maximising output on the current sales of its rival. They can be illustrated by q1 and q2 as in Fig. 12.2. It is seen that the reaction curves have a negative slope. They are also linear in our specific example, although this is not necessary.
The equilibrium solution is derived from a simultaneous solution of the reaction curves. This is shown by point E. At E, the sales decision of each firm corresponds to the expectations on which the other bases its calculations. Hence the industry is in equilibrium at E.
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.
Illustration – Mineral Spring-Water Case:
We can illustrate the above conclusions with Cournot’s simple case of mineral spring water. Here the costs are same for both firms, and are identically zero (c = 0). This means that the supply of both firms will be equal (q1 = q2)
Since, Ci = 0 and q1 = q2, the reaction curves eqns. [4] and [5] become –
This implies –
qi = (a/3b)
The industry’s output and price in Cournot equilibrium are –
Under simple monopoly MR = MC. Since MC = 0, MR = 0. Hence, MR = 0 = a – 2bq.
This implies that –
Thus, it can be seen that monopoly output is less than, and monopoly price larger than that under Cournot duopoly.
Since marginal cost is zero in perfect competition, indefinitely large quantities of spring water will be supplied at any positive price, driving the price down to zero. The competitive output will be equal to the demand at zero price.
p = 0 = a – bQ
which implies
Q = a/b
We thus see that the competitive output is 3/2 times as large as the output in Cournot’s duopoly. By analogous reasoning, we can show that the combined profits of duopolists will be positive and smaller than the profits of the monopoly, and larger than under competition.
Generalization to ‘n’ Sellers:
The Cournot duopoly results can be generalised to an oligopoly with n sellers. In this general case, assuming constant average and marginal costs d, we can show the industry’s output Q and price p to be –
Cournot and Other Market Forms:
The general Cournot model yields the industry results of other structures as special cases. Thus,
(i) n = 1, gives the monopoly solution;
(ii) n = 2, gives the duopoly solution;
(iii) n → ∞, gives the competitive solution; and
(iv) At n, the Cournot oligopoly output is n/(n+1) times the competitive output and the oligopoly price is higher than the competitive price.
Thus, pure monopoly and perfect competition are revealed to be limiting forms of the generalized Cournot oligopoly.
Limitations:
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. 12.3. 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 MCA= MCB = 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 pc 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. 12.3, the competitive price is represented by pc. 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 pc, 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:
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. 12.4. 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.
The Kinked Demand Hypothesis:
Sweezy suggests that an oligopolist’s demand curve appears to be kinked as in Fig. 12.5. The kink in the demand curve reflects an asymmetry in the elasticity of demand at the going price. Thus, demand is less elastic for a price cut at p, but more elastic for a price increase at the same price p.
This means that the marginal revenue will be less below p, and higher above. Hence there appears a gap in the marginal revenue curve at the going level of sales as shown in the Fig. 12.5.
Explanation:
Why should the demand curve appear kinked? The kink appears because rivals react asymmetrically to a change in the price by the firm, and because the rival’s products are substitutes for the firm’s products. Hence the oligopolist conjectures that a price cut will be followed by all rivals because they fear that they may lose their share in the market if they do not. In contrast, a price hike by the oligopolist will find no firms following suit, since they do not feel that a price increase by their rival can threaten their market share. This is the asymmetry in the rivals’ reactions to the price variations by the firms.
Rivals’ reactions could have had no effect on the firm’s demand curve were it not for the fact that their products are substitutes. Since the products are substitutes, a part of the oligopolist’s demand switches over to his rivals when he raises his price but his rivals do not. This explains the higher elasticity of the demand curve above the going price. A price decrease however fails to draw demand away from the rivals because all of them follow suit. Hence the demand is less elastic for a price cut.
Significance:
The kinked demand hypothesis explains price rigidity in non-collusive oligopolies. The firm does not increase its price if its costs increase within ‘tolerable’ limits, since its demand is elastic above the going price. On the other hand, lowering the price in response to lower costs is not very rewarding because demand is less elastic for price cuts. Hence ‘tolerable’ changes in costs are absorbed by the firm which maintains its price and sales level.
What are these ‘tolerable’ limits to changes in costs? The ‘tolerable’ limits are indicated by the gap in the firm’s marginal revenue curve at the current level of sales in Fig. 12.5. This gap in the marginal revenue absorbs the ‘shocks’ of changes in costs. Only if the marginal costs increase or fall sufficiently to spill out of this gap, will the firm change its price and output.
Conclusions of the Model:
Sweezy’s model explains the rigidity of the going price of product. It does not explain how this price is arrived at. Thus it is not a theory of price determination as was thought when it was first proposed.
The model predicts price rigidity where there is asymmetric rival reaction, and the products of the rivals are substitutes. By inference, the more the product heterogeneity, the lesser is the price rigidity. Further, price rigidity can be expected to be less when oligopolists collude in price formation so that the prices tend to move in phase (upward or downwards). Finally, the price rigidity may also be expected to be less if there are no rivals, that is, when the firm is a monopoly.
Empirical Evidence:
Initial evidence for the kinked demand model seemed to be favourable. For instance, in the US potash industry in 1934, one major firm raised its price without its rivals following suit, with the result that the firm’s sales fell off dramatically. As a result, it subsequently cut its prices below the level prevailing in the rest of the industry. The reluctance to follow price increases revealed by this example may have been due to the glut in the markets during the 30’s. In fact, Profs. Stonier and Hague opine that the kinked demand curve is likely to be found mainly where trade is relatively depressed.
Later studies by Stigler did not reveal any asymmetry in rival’s reactions to price changes by a single firm. Nor was there any evidence to show that price rigidity was more in non-collusive oligopolies when compared with monopolies, or when compared with the same markets in periods of known collusion. Finally Stigler could not find any evidence to show that price rigidity increased with the homogeneity of the products.
A still later study by Primeaux and Bomball found that during 1959-62 and 64-70:
(a) Price decreases were not followed any more frequently than price increases;
(b) Price increases were more nearly simultaneous than were price decreases;
(c) There was significantly more price rigidity in monopoly market structures than in oligopoly market structures.
These and other empirical findings have raised doubts over the general validity of the kinked demand curve model.
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