What is conjectural variation? Show how some models of oligopoly are derived on the basis of it.

Understanding Conjectural Variation

Conjectural variation (CV) represents a firm’s belief about how much its rivals’ output will change in response to a change in its own output. It is mathematically expressed as the change in the rival’s output divided by the change in the firm’s output. CV can range from 0 to 1, and even be negative.

• CV = 0: The firm believes its rivals will not change their output regardless of its actions (monopoly situation).
• 0 < CV < 1: The firm believes its rivals will increase output, but by less than the increase in its own output. This is typical of Cournot competition.
• CV = 1: The firm believes its rivals will match any change in its output (Bertrand competition).
• CV > 1: The firm believes its rivals will increase output by more than its own increase (cooperative or collusive behavior).
• CV < 0: The firm believes its rivals will decrease output if it increases its own (rare, but possible in certain scenarios).

Oligopoly Models and Conjectural Variation

1. Cournot Model

The Cournot model, developed by Antoine Augustin Cournot in 1838, assumes that firms compete by choosing quantities simultaneously. The key assumption regarding conjectural variation in the Cournot model is 0 < CV < 1. Each firm believes that its rivals will maintain their output levels if it changes its own, or increase output by less than the amount of its increase.

The reaction function for each firm in a duopoly can be derived as follows:

Let Q = q1 + q2 be the total market output, and P(Q) be the inverse demand function. Firm 1 maximizes its profit:

π1 = P(Q) * q1 – C1(q1)

Taking the first-order condition and assuming CV = α (where 0 < α < 1), we get:

∂π1/∂q1 = P(Q) + q1 * ∂P/∂Q + α * q2 * ∂P/∂Q = 0

This equation defines Firm 1’s reaction function, showing how its optimal output (q1) depends on its belief about Firm 2’s output (q2) and the value of α.

2. Bertrand Model

The Bertrand model, proposed by Joseph Bertrand in 1883, assumes that firms compete by setting prices simultaneously. The crucial assumption here is CV = 1. Each firm believes that if it lowers its price, its rivals will immediately match that price reduction. This leads to a highly competitive outcome, often approaching perfect competition.

In the Bertrand model, if one firm lowers its price, the other firm will match it, driving the price down to marginal cost. This is because any price above marginal cost will be undercut by the rival, leading to zero profits for the firm charging the higher price. The Nash equilibrium in the Bertrand model is where both firms price at marginal cost.

3. Stackelberg Model

The Stackelberg model, developed by Heinrich von Stackelberg in 1934, introduces the concept of a leader and a follower. The leader firm chooses its output first, and the follower firm then chooses its output, taking the leader’s output as given. In this model, the leader firm considers the follower’s reaction function when making its own output decision. The conjectural variation for the leader is different from that of the follower.

The leader firm assumes that the follower will react according to its reaction function. The leader incorporates this reaction into its own profit maximization problem. The follower, on the other hand, assumes a CV of 0, believing the leader’s output is fixed when it makes its own output decision.

This leads to a situation where the leader firm has a first-mover advantage and can achieve a higher profit than the follower.

Limitations of Conjectural Variation

While a powerful tool, the concept of conjectural variation has limitations. It relies on subjective beliefs about rivals’ behavior, which may not always be accurate. Furthermore, it doesn’t explain *how* these beliefs are formed. Game theory, with concepts like repeated games and signaling, provides more sophisticated frameworks for analyzing strategic interactions in oligopolistic markets.