V. Un consumidor inerte
3.6 Los profesionales del futuro
Stackelberg (1934) suggested yet another solution to the Cournot duopoly model.
The Cournot model assigns equal status to both firms as they progress towards the final equilibrium. Both firms operate according to the zero conjectural variation assumption, and each firm fails to anticipate the other’s reaction on each occasion it adjusts its own output. Suppose, however, we drop the zero conjectural variation assumption for firm A, but retain this assumption for firm B. B continues to select its profit-maximizing output by treating A’s output as fixed at its current level. But A learns to recognize that B behaves in this manner. A therefore learns to take B’s behaviour into account whenever A makes its own output decisions.
How should firm A select its own output, given that it has this insight into firm B’s behaviour? A’s awareness of B’s behaviour is tantamount to A’s recognition that whatever output A selects, B always reacts by selecting an output that returns the two firms to an output combination that lies on B’s reaction function, RFB. A should therefore select the output that maximizes A’s profit subject to B’s expected reaction. Accordingly, A should select qA
L and aim for SAin Figure 4.8: the point on RFBwhere A’s profit is maximized. A anticipates, correctly, that B will react by pro-ducing qBF. SAis the point of tangency between RFBand the highest isoprofit curve A can attain, given that the final equilibrium must lie on RFB. At any other point on RFB, A’s profit is lower than it is at SA.
By learning to anticipate and take account of firm B’s behaviour, firm A earns a higher profit than at C–N, while B earns a lower profit. A is rewarded, and B is punished, for the fact that A has insight into B’s behaviour, while B does not have corresponding insight into A’s behaviour. An alternative (but only slightly different) interpretation of Stackelberg’s solution is as a model of first-mover advantage.
Returning to Cournot’s original story of sequential decision making, if A recognizes that B always follows the zero conjectural variation assumption, in Round 1 A should produce qA
L, in the knowledge that B will react by producing qB
F. Accordingly, the two firms arrive directly at the Stackelberg equilibrium at the end of Round 1, with A producing the higher output and earning the higher profit. In this interpretation, A is the leader and B is the follower, and A is rewarded for its first-mover advantage.
130 Chapter 4 n Oligopoly: non-collusive models
Generalizing the preceding discussion, we can identify four possible outcomes, as shown in Figure 4.8:
n At SA, firm A is the leader and firm B is the follower, as discussed above.
n SBrepresents the opposite case, where B is the leader and A is the follower. A follows the zero conjectural variation assumption. B recognizes A behaves in this way, and aims for SB, the point on A’s reaction function RFAthat maximizes B’s profit.
n If both firms are followers, C–N, the Cournot–Nash equilibrium, is achieved as before.
n Finally, and quite realistically in many oligopolistic markets, both firms might similtaneously attempt to be leaders. If both simultaneously produce the higher level of output qA
L = qB
L, the result is a Stackelberg disequilibrium or price war at P–W. At this conflict point there is overproduction, and the firms are forced to cut their prices in order to sell the additional output. Accordingly, both firms earn less profit than at C–N. A costly price war might eventually determine a winner and a loser, but it is also possible the firms may realize the futility of conflict and search for a more cooperative solution.
In Section 4.3, we have identified a number of possible solutions to the problem of output determination in duopoly. A mathematical derivation of these results can be found in Appendix 1. To conclude this section, it is useful to return to the numerical example that was used to introduce the Cournot model at the start of this section, and compare the numerical values of price and quantity for each of the solutions to the model. Accordingly, we now consider a duopoly in which the market demand
4.3 Models of output determination in duopoly 131
Figure 4.8 Cournot–Nash equilibrium and Stackelberg equilibria
function is linear and the units of measurement for price and quantity are scaled from 0 to 1; and both firms produce at zero marginal cost. Figure 4.9 shows the numerical values of qA and qB at the Cournot, Chamberlin and Stackelberg equilibria. The following table contains the same numerical data, and also compares the equilibrium prices and profits of the two firms.
P Q qA qB ππA ππB
Cournot–Nash 1/3 2/3 1/3 1/3 1/9 1/9
Chamberlin 1/2 1/2 1/4 1/4 1/8 1/8
Stackelberg – A as leader 1/4 3/4 1/2 1/4 1/8 1/16
Stackelberg – B as leader 1/4 3/4 1/4 1/2 1/16 1/8
Stackelberg disequilibrium (price war) 0 1 1/2 1/2 0 0
The Chamberlin joint profit maximization equilibrium corresponds to the mono-poly price and output, with the firms sharing the monomono-poly profit equally between them, with πA= πB= 1/8. Both are better off than at the Cournot–Nash equilibrium, where price is lower, total output is higher, and πA= πB= 1/9. At the Stackelberg equilibrium with A as leader, price is lower still, and total output is higher. A does better (πA= 1/8) and B does worse (πB= 1/16) than at the Cournot–Nash equilibrium.
At the Stackelberg equilibrium with B as leader, these positions are reversed. Finally, the Stackelberg disequilibrium (price war) corresponds to the perfectly competitive price and output, with price driven down to zero (equal to marginal cost), output raised to one, and both firms earning zero profit.
132 Chapter 4 n Oligopoly: non-collusive models
Figure 4.9 Equilibrium values of qA, qB, πA, πB: duopoly with linear market demand and zero marginal cost