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The most interesting set of assumptions about conjectural variation has been made in the analysis of leadership and followership formulated by the German economist Heinrich von Stackelberg. The Stackelberg model is a quantity leadership model. It describes the strategic behaviour of industries in which there is a dominant firm or a natural leader and the other firms are the followers. For simplicity here we consider as duopoly situation, as in Cournot’s model.

**In general, the profit of each duopolist is a function of the output levels of both:**

π_{1} = f_{1}(q_{1}, q_{2}), π_{2} = f_{2}(q_{1}, q_{2}) ……………. (1)

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The Cournot solution is obtained by maximising π_{1} with respect to q_{1}, assuming q_{2} to be constant, and with respect to q_{2}, assuming q_{1} to be constant. Thus, each firm might make the same assumption about its rival’s response.

**Profit maximisation by the two duopolists then requires:**

The terms ∂q_{2}/∂q_{1} and ∂q_{1}/∂q_{2} represent conjectural variation, i.e., the assumed response of each firm to its rival’s output. If firms make any wrong assumptions about each other’s response then equation (2) will not represent an improvement over the Cournot model.

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In this model a follower obeys his reaction function q_{2} = f_{2} (q_{1}) and adjusts his output level to maximise his profit, given the quantity decision of his rival, who he assumes to be a leader. A leader does not obey his reaction function q_{1} = f_{1}(q_{2}). He assumes that his rival acts as a follower and maximises his profit, given his rival’s reaction function.

**If firm 1 desires to act as a leader, he assumes that firm 2’s reaction function is valid and substitutes this reaction into his profit function:**

π_{1} = f_{1} [q_{1}, f_{1}(q_{1})] ……………. (3)

Firm l’s profit is now a function of q_{1} alone and can be maximised with respect to this single variable.

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Let us suppose firm 1 is the leader and it chooses to produce a quantity q_{1}. Firm 2 responds by choosing a quantity q_{2}. Each firm is aware of the fact that equilibrium price in the market depends on total output produced. We use the inverse demand function p(Q) to indicate the equilibrium price as a function of industry output, Q = q_{1} + q_{2} or p(Q) = p(q_{1} + q_{2}), since P = f(Q) or p = p(Q).

The leader’s output choice depends on how the leader thinks that the follower will react to its choice. In all likelihood the leader should expect to follow in an attempt to maximise profits as well, given the choice made by the leader. In order for the leader to make a rational decision about its own production, it has to consider the follower’s profit-maximisation problem.

**The Follower’s Problem:**

**The Stackelberg model assumes that the follower wants to maximise its profits:**

The follower’s profits depends on the output choice of the leader, but from the follower’s viewpoint, the leader’s output is a predetermined variable. The leader’s output has already been produced. So the follower just takes it as a constant.

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**The follower wants to choose an output level such that MR equals MC:**

MR_{2} = p_{1}(q_{1} + q_{2}) + Δp/Δq_{2}.q_{2} = MC_{2} …………… (5)

The MR is less than p as is the case with any non-purely competitive market. When the follower increases its output, it earns more revenue by selling more at the market price. But it has also to reduce the price by Ap. This lowers its profits on all the units that could previously be sold at the higher price.

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An important point made by the Stackleberg model is that the profit-maximising choice of the follower will depend on the choice made by the leader.

**This relationship can be expressed as: **

q_{2} = f_{2}(q_{1}) ……………. (6)

This function f_{2}(q_{1}) indicates the profit-maximising output of the follower as a function of the leader’s choice. This function is called the reaction function since it shows how the follower will react to the leader’s output decision.

**Derivation of the Reaction Curve:**

In case of linear demand the inverse demand function takes the form p(q_{1} + q_{2}) = a – b(q_{1} + q_{2}). For the sake of simplicity costs are assumed to be zero.

Then the profit function for firm 2 (follower) is;

π_{2}(q_{1}, q_{2}) = Pq_{2}, since c_{2} = 0

= [a – b (q_{1} + q_{2})] q_{2}

or, π_{2}(q_{1}, q_{2}) = aq_{2} – bq_{1}q_{2} – bq_{2}^{2} ……………….. (7)

**Iso-Profit Curves: **

We use equation (7) to derive the iso-profit curves in Fig. 24.4. These are lines showing those combinations of q_{1} and q_{2} that yield the same level of profit to firm 2, such as π_{2}.

**In other words, the iso-profit curves are constructed by connecting all points (q _{1}, q_{2}) that satisfy equations of the form: **

Aq_{2} – bq_{1}q_{2} – bq_{2}^{2} = π_{2} … … (8)

**Reaction Curve of Firm 2: **

Firm 2’s profits will increase as it moves to iso-profit curves that are further to the left. This is because if firm 2 fixes its output at some level, its profits will increase if and only if firm 1’s output falls. Firm 2 will be able to make the maximum amount of profit when it is a monopolist, that is, when firm 1 decides not to produce any output.

For each possible choice of firm l’s output, firm 2 will choose that level of output which enables it to make the maximum amount of profits. This implies that for each level of q_{1}, firm 2 will choose the value of q_{2} which enables it to move the iso-profit curve furthest to the left, as shown in Fig. 24.4. Point E gives the maximum profit to firm 2. At this point the slope of the iso-profit curve is zero.

The locus of all the tangency points such as E, F, G and H is the reaction curve of firm 2, i.e., f_{2}(q_{1}). The reaction curve depicts the profit- maximising output for the follower, firm 2, for each output choice of the leader, firm 1. For each choice of q_{t} the follower chooses the output level f_{2}(q_{1}) associated with the iso-profit line furthest to the left.

**The marginal revenue associated with the profit function for firm 2 is shown by the following equation:**

MR_{2 }(q_{1}, q_{2}) = a – bq_{1} – 2bq_{2}

**Since marginal cost is zero in this model, if we set the marginal revenue equal to marginal cost, we get:**

a — bq_{1} — 2bq_{2} = 0 …….. (9)

**Which, when solved, gives us the following equation of firm 2’s reaction curve:**

2bq_{2} = a – bq_{1 }

Or, q_{2} = (a – bq_{1})/2b ……………… (10)

**The Leader’s Problem:**

Like the follower, the leader is also aware that its output decisions influence the output choice of the follower. This relationship is captured by the reaction function f_{2} (q_{1}). Thus while taking its output decision, it should recognise the influence it exerts on the follower.

The profit-maximising problem for the leader, therefore, becomes;

Max _{q1}. p (q_{1 }+q_{2}) q_{1 }– c_{1}(q_{1})

such that q_{2} = f_{2 }(q_{1})

If we substitute the second equation into the problem it becomes;

Max _{q1}. p [p_{1} + f_{2}(q_{2})]q_{1} – c_{1}(q_{1})

The leader recognises that when it chooses output q_{1}, the total output produced will be q_{1} +f_{2 }(q_{1}): its own output plus that of the follower.

When the leader plans to change its output it has to recognise the influence it exerts on the follower. The reaction function of firm 2 (the follower) is given by;

f_{2} (q_{1}) = q_{2} = a – bq_{1}/2b … … (11)

**Here the leader’s profits are: **

π_{1} (q_{1}, q_{2}) = p_{1} (q_{1} + q_{2})q_{1} = aq_{1} – bq_{1}^{2} – bq_{1}q_{2} … … (12)

since marginal costs are zero and p(q_{1} + q_{2}) = a – b(q_{1} + q_{2}) in case of a linear inverse demand function.

Thus total profit is the same as total revenue because total cost is zero. This means that profit maximisation is equivalent to revenue maximisation. But the output choice of the follower, q_{2}, will depend on the leader’s choice via the reaction function q_{2} – f_{2 }(q_{1}).

Substituting from equation (11) into equation (12) we have;

**Graphical Illustration of the Model****:**

The Stackelberg model is graphically illustrated in Fig. 24.5 by using the iso-profit curves of firms only, which are a type of indifference curves. But we draw the reaction curves of both firms. Only higher profits for firm 1 are associated with iso-profit curves which are lower down since firm 1’s profits will increase as firm 2’s output falls.

Being a follower, firm 2 will choose an output along its reaction curve, f_{2}(q_{1}). Thus firm 1 wants to choose an output combination on firm 2’s reaction curve which gives it the maximum amount of profit. But the maximum amount of profits means choosing that point on firm 2’s reaction curve that touches firm l’s lowest attainable iso-profit curve, as shown by point E in Fig. 24.5. The standard rule of optimisation (in this case profit maximisation) suggests that the reaction curve must be tangent to the iso-profit curve at this point.

**Comparison with Cournot Model: **

In Fig. 24.5 we also show Cournot equilibrium point c, where the two reaction curves meet. While the Cournot model is one of simultaneous quantity setting, the Stackelberg model a quantity leadership model.

Stackelberg equilibrium is attained if and only if firm 1 desires to be a leader and firm 2 a follower. Under the Stackelberg assumptions, the Cournot solution is achieved if each desires to act as a follower, knowing that the other will also act as a follower. Otherwise, one must change his pattern of behaviour and act as a leader before equilibrium can be attained.

**Stackelberg Disequilibrium:**

If both firms desire to be leaders, each assumes that the other’s behaviour is governed by its reaction function, but, in fact, neither of the reaction functions is obeyed. As a result a Stackelberg disequilibrium situation is created. Stackelberg believed that this disequilibrium situation is encountered more often than not.

So it is not possible to produce the final result of a Stackelberg disequilibrium on a priori basis. If Stackelberg was correct. Such on this situation would lead to a some sort of price war and equilibrium will not be attained until one has succumbed to the leadership of the other or a collusive agreement has been reached. In the Stackelberg model each duopolist makes greater profit by being able to act as a leader. So both desire to act as leaders. This point may now be discussed.

**First-Mover Advantage: **

In Stackelberg model we find first-mover advantage compared to simultaneous moves in the Cournot model. The Stackelberg model is about strategic competition. In a Cournot model firm 1 would take firm 2’s output as fixed and given. The Stackelberg model highlighted the value of (extra) information and the potential value of being a market leader, in the sense of being able to act first in setting output.

Both insights were derived by noting that the added information about how firm 1 would sights were derived by noting that the added information about how firm 1 would behave made firm 1 a ‘Stackelberg leader’ that enjoyed a strategic advantage over firm 2, the ‘Stackelberg follower’. The Stackelberg model is based on the assumption that firm 1 knows as much about firm 2’s reaction function as anyone else in the market and can use that information in its own output determination.

A Stackelberg equilibrium would not occur at the point where the two firms reaction (or best response) curve’s intersect because firm 1 would no longer take firm 2’s output as fixed. On the contrary it would recognise that firm 2’s output would depend on its own and this recognition would allow it to raise its profits above the level sustained by a Cournot equilibrium.

To maximise profit in the Stackelberg model, firm 1 would, therefore, choose to produce where MR exactly matched MC. More generally, the Stackelberg leader would expand its output relative to the Cournot equilibrium at the expense of a reduction in the output of the follower.

And it would certainly be a non-cooperative (non-collusive) Nash equilibrium. Firm 1 would be maximising profit given that firm 2 was behaving as a Cournot duopolist. It would, therefore, have no incentive to change anything because it will gain by taking a lead.

There is an obvious gain from^{ }being a market leader and being able to ‘move first’. Suppose that firm 1 got to choose its output level with a complete information on how firm 2 would respond. In Fig. 24.6 firm 1 would maximise its profit by choosing output p’_{1} = 150 units which would be larger than the Cournot equilibrium output of q_{1} = 100 units.

Moving first would therefore, give firm 1 a strategic advantage over firm 2. It could establish its own output level and firm 2 would be left to react as best as it could (maximizing its own profit by settling for the relatively low output level of q’_{2} = 75 units < 100 units). The advantage that firm 1 enjoyed by going first is often called the ‘first-mover advantage’. Thus the Stackelberg leader (firm 1) produces more output than it would under Cournot equilibrium while the Stackelberg follower (firm 2) produces less.

**An Alternative Interpretation: **

In Fig. 24.7 the residual demand curve the Stackelberg leader faces is the market demand Q minus the quantity produced by the follower, Q_{F}, given the leader’s quantity Q_{L}. The leader chooses Q_{L} so that its MR curve in part (a) intersects its MC curve at point E. The total output Q is the sum of the output of the two firms.

In part (b) (small) the quantity the follower produces is its best response to the leader’s output, as given by its Cournot best-response (or reaction) curve. If market demand is 144, firm 1 produces 96 (point E’), and firm 2 will produce 48. If market demand is 192, and firm 1 produces 96, firm 2 will also produce 96 (Q’_{F}).

**Comparison with the Cournot Model****: **

The Cournot and Stackelberg models are alternative ways of representing oligopolistic behaviour. The Stackelberg model is different from the Cournot model, in which neither firm has any opportunity to react. The Cournot model is applicable to industries in which all firms are similar and none has a strong operating advantage or leadership position. By contrast, the Stackelberg model is appropriate (more realistic) in case of industries in which a single firm occupies the dominant position and the firm usually takes the lead in introducing new products or setting price.

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