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Chapter 5: Cartels: Oligopoly Joint Decision Making |
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Who is so deafe or so blinde as is heeCournot's model was the first conjectural variation model (and first oligopoly model). In Cournot's original story, a firm makes one of the simplest possible assumptions about the behavior of other firms: other firms continue to produce the same level of output no matter how it behaves. That is, each firm's conjecture is that other firms are satisfied to continue selling their current quantity of output. This very parsimonious and strong assumption leads to clear behavioral implications, but it is arbitrary and may be incorrect.
That wilfully will neither heare nor see? — JOHN HAYWOOD
Stackelberg follower firms make the Cournot conjecture. The Stackelberg leader takes those conjectures into account in making its decisions. In the Bertrand model, each firm makes the conjecture that its rivals will not change their price in response to a change in its own price.
Second, and perhaps more devastating, multiperiod interpretations of conjectural variations models are implausible. Unlike multiperiod game theory models, which have equilibria based on credible strategies that do not contradict firms' beliefs, dynamic conjectural variations models are based on inconsistent beliefs and actions.
Consider a standard multiperiod interpretation of the Cournot model of the melons market discussed in the text. Suppose, in the first period, Firm 1 produces 300 melons and Firm 2 produces 200. Then, in Period 2, Firm 1 uses its (one period) best-response function and produces R1(200) = 360 - q2/2 = 260 melons. Firm 2 produces R2(300) = 360 - q1/2 = 210 melons in the second period. In subsequent periods, each firm continues to use its one-period best-response function to react to its rival's output in the previous period. Eventually, the output of each firm converges to the Cournot equilibrium of 240 melons.
There are two serious problems with this multiperiod interpretation of a Cournot model. First, the firms' myopia over time is unreasonable. A firm can observe the response of its rivals to changes in its output during the last period and can verify that those changes do affect what its rivals do. In other words, although a firm sees that its Cournot conjecture of no response by its rival is wrong, it continues to rely on this false conjecture. That is, it continues to use its one-period best-response function. Alternatively stated, the Cournot equilibrium is not robust with respect to experimentation. At the equilibrium, if one firm varies its output slightly, as an experiment, and observes how the other firm behaves, it will learn that its Cournot assumption of no reaction is false.
Second, in a multiperiod model, it does not make sense for a firm to maximize its profits in the current period alone, as these models imply. Rather, the firm should maximize the present discounted value of its future stream of profits. A firm may be willing to trade large profits in the future for smaller profits today. Indeed, firms can increase profits by varying their behavior over time as shown in the section on multiperiod games.
Equation 1
1 = q1p(q1 + q2) - C(q1),
through its choice of its output level, q1, where industry price, p, is a function of industry output and its cost function, C(·), is the same as for Firm 2. Firm 1's first-order condition is
Equation 2
(d1/dq1) =
p + q1p'(q1 + q2)(1 +
(dq2/dq1)) - C'(q1) = 0,
where C'(q1) is its marginal cost and p'(q1 + q2) is the slope of the inverse demand function. Firm 1 holds a conjecture v = dq2/dq1 about the other firm, so Equation (2) can be written as
Equation 3
p + q1p'(q1 + q2)(1 + v) - C'(q1) = 0.
By symmetry, the first-order condition for Firm 2 is the same, with the 1 and 2 subscripts reversed.
If v is 0, Equation 3 is the first order condition for a Cournot firm. That is, a Cournot firm conjectures that the other firm will not change its output in response to a change in its own output level. Setting v = 0 in Equation 3 gives the usual Cournot first-order condition.
If v = -1, the firm holds the Bertrand competitive conjecture (Telser 1972). A Bertrand firm believes that any increase in its output is exactly offset by a decrease of its rivals' output, so that the market price remains unchanged. If there are n identical firms, the Bertrand conjecture is v = -1/(n - 1). Substituting v = -1 into Equation 3 gives the equilibrium condition p = C' (price equals marginal cost), which is the Bertrand or competitive equilibrium.
If v = 1, the firm holds a conjecture that leads to the collusive equilibrium if firms behave symmetrically (q1 = q2). The firm believes that its rival will change its output by the same amount as it does. Thus, the firm can affect total industry output, but not its market share (of 50 percent) by varying its output. The firm cannot increase its profits at the expense of the other firm, so it produces the cartel output. The first-order condition becomes p + 2q1p'(Q) = C'(q1), which is the cartel solution because total output, Q = q1 + q2 = 2q1 if q1 = q2.
Other values for v are also possible. If one estimates v (see Chapter 9), using Equation 3, one can calculate Lerner's Index, (p - C')/p. Thus, one can interpret v as either a conjectural variation or as a measure of the gap between price and marginal cost.
Bowley, Arthur L. 1924. The Mathematical Groundwork of Economics. Oxford: Oxford University Press.
Stigler, George J. 1947. "The Kinky Oligopoly Demand Curve and Rigid Prices." Journal of Political Economy. 55:432-49.
------. 1964. "A Theory of Oligopoly ." Journal of Political Economy 72:55-59.
Tesler, Lester G. 1972. Competition, Collusion and Game Theory. Chicago: Aldine-Atherton.
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