The main problem associated with implicitly collusive pricing is that it is difficult for firms to agree (without communicating with one another) on what the appropriate price should be. There is often co-ordination failure when cost and demand conditions and thus the ‘correct’ price keeps on changing. Price signaling is one form of implicit collusion that seeks to tackle the problem.
For example, a firm might announce through newspapers and TV advertisements that it has raised its price with the expectation that its competitors will take this announcement as a signal that they should also raise prices. If competitors follow suit, a win-win situation is likely to be created, in which case all of the firms will earn higher profits at least in the short run.
At times a different type of behaviour is established whereby one firm regularly announces price changes and other firms in the industry follow suit. This pattern of behaviour is called price leadership. One firm is implicitly recognized as leader by all other firms, the followers. The followers match the leader’s prices. In short, price leadership solves the problem of coordinating price: each follower charges exactly what the leader is charging.
Price leadership serves another important purpose. It enables oligopolistic firms to deal with the reluctance to change prices, which arises from the apprehension of being undercut. With changes in cost and demand conditions, firms may find it increasingly necessary to change prices that have remained rigid for quite some time. In such a situation, followers might look to a price leader to signal when and by how much price should change. Sometimes a large dominant firm will act as a natural leader. At other times, different firms will act as leaders from time to time.
Leadership models of oligopoly are of two types — price leadership model and quantity leadership model. A leader may set quantity. Alternatively, he may set price. In order to make a rational pricing decision, the leader has to forecast the behaviour of the follower.
It may be noted at the outset that since two firms are assumed to sell identical products the follower must always set the same price as that of the leader. If firm 1 charged a price which was marginally higher than that of firm 2, firm 1 would lose all its customers to firm 2. In such a situation, it is not possible to think of an equilibrium with both firms producing a positive quantity.
Let us suppose the leader has set the price at p. We also suppose that the follower acts as a price taker. It takes this price as given and deter- mines its optimum (profit-maximising) output. This is the same as the behaviour of a price-taking firm under perfect competition. In this case, price is outside the control of the follower because it has already been set by the leader.
The follower seeks to maximise profits:
This means that the follower will want to choose an output level where price equals marginal cost. This condition enables us to determine a supply curve for the follower, S(p) as shown in Fig. 24.8. Now let us suppose the leader realizes that if he sets a price p, the follower will supply S(p).
So the leader will be able to sell the difference between market demand D(p) and the quantity supplied by the follower S(p). The difference between the two is known as residual demand R(p) = D(p) – S(p). This is indicated by the demand curve faced by the leader called the residual demand curve DL[R(p)].
Let us suppose the marginal cost of production of the leader is constant at c.
Then its profits at the price p are:
π1 (p) = (p – c) [D(p) – S(p)] = (p – c) R(p)
In order to maximise profits the leader has to choose that particular price-quantity combination which equates marginal revenue with the marginal cost. However, the marginal revenue curve is the one that corresponds to the residual demand curve which measures how much output the leader will be able to sell at each price. Since the residual demand curve is linear, the corresponding marginal revenue curve will have the same intercept on the vertical axis but double the slope.
The leader equates marginal revenue and marginal cost to find the optimal quantity, q1* (corresponding to point E). The total quantity offered for sale in the market is qT* (corresponding to point F) and the equilibrium price is p*. So the follower sells q1* qT* units at this price.
The Algebra of Price Leadership:
Let us suppose the inverse demand function is D(p) – a – bp. The cost function of the follower is assumed to be C2(q2) = q22/2, and that of the leader is c1(q1) = cq1.
For any price p the follower seeks to operate where p = MC. Its marginal cost function is MC2(q2) = q2. By setting price equal to marginal cost we get;
p = q2
And if we solve the follower’s supply function we get q2 = S(p) = p.
The residual demand function is R(p) = D(p) – S(p) = a – bp – p = a – (b + 1)p. Solving for p as a function of the leader’s output q1, we get;
q1 = a – (b + 1 )p
or (b + 1)p = a – q1
or p = a/b + 1 – q1/b + 1 = a/b + 1 – 1/b + 1 q1
This is the inverse demand function faced by the leader.
The corresponding marginal revenue function is: