You are reading “Market Profile Theorems”, one entry from Oasis Market
- Published:
- 23.04.10 / 6pm
Market Profile Theorems
The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. It is named after the German economist Heinrich Freiherr von Stackelberg who published Market Structure and Equilibrium (Marktform und Gleichgewicht) in 1934 which described the model.
In game theory terms, the players of this game are a leader and a follower and they compete on quantity. The Stackelberg leader is sometimes referred to as the Market Leader.
There are some further constraints upon the sustaining of a Stackelberg equilibrium. The leader must know ex ante that the follower observes his action. The follower must have no means of committing to a future non-Stackelberg follower action and the leader must know this. Indeed, if the 'follower' could commit to a Stackelberg leader action and the 'leader' knew this, the leader's best response would be to play a Stackelberg follower action.
Firms may engage in Stackelberg competition if one has some sort of advantage enabling it to move first. More generally, the leader must have commitment power. Moving observably first is the most obvious means of commitment: once the leader has made its move, it cannot undo it - it is committed to that action. Moving first may be possible if the leader was the incumbent monopoly of the industry and the follower is a new entrant. Holding excess capacity is another means of commitment.
Subgame perfect Nash equilibrium
The Stackelberg model can be solved to find the subgame perfect Nash equilibrium or equilibria (SPNE), i.e. the strategy profile that serves best each player, given the strategies of the other player and that entails every player playing in a Nash equilibrium in every subgame.
In very general terms, let the price function for the (duopoly) industry be P ; price is simply a function of total (industry) output, so is P ( q 1 + q 2 ) where the subscript 1 represents the leader and 2 represents the follower. Suppose firm i has the cost structure C i ( q i ) . The model is solved by backward induction. The leader considers what the best response of the follower is, i.e. how it will respond once it has observed the quantity of the leader. The leader then picks a quantity that maximises its payoff, anticipating the predicted response of the follower. The follower actually observes this and in equilibrium picks the expected quantity as a response.
To calculate the SPNE, the best response functions of the follower must first be calculated (calculation moves 'backwards' because of backward induction).
The profit of firm 2 (the follower) is revenue less cost. Revenue is the product of price and quantity and cost is given by the firm's cost structure, so profit is:
. The best response is to find the value of
q
2
that maximises
Π
2
given
q
1
, i.e. given the output of the leader (firm 1), the output that maximises the follower's profit is found. Hence, the maximum of
Π
2
with respect to
q
2
is to be found. First differentiate
Π
2
with respect to
q
2
:
Setting this to zero for maximisation:
The values of q 2 that satisfy this equation are the best responses. Now the best response function of the leader is considered. This function is calculated by considering the follower's output as a function of the leader's output, as just computed.
The profit of firm 1 (the leader) is Π 1 = P ( q 1 + q 2 ( q 1 )). q 1 − C 1 ( q 1 ) , where q 2 ( q 1 ) is the follower's quantity as a function of quantity, namely the function calculated above. The best response is to find the value of q 1 that maximises Π 1 given q 2 ( q 1 ) , i.e. given the best response function of the follower (firm 2), the output that maximises the leader's profit is found. Hence, the maximum of Π 1 with respect to q 1 is to be found. First derive Π 1 with respect to q 1 :
Setting this to zero for maximisation:
Examples
The following example is very general. It assumes a generalised linear demand structure
and imposes some restrictions on cost structures for simplicity's sake so the problem can be resolved.
for ease of computation.
The follower's profit is:
The maximisation problem resolves to (from the general case):
Consider the leader's problem:
Substituting for q 2 ( q 1 ) from the follower's problem:
The maximisation problem resolves to (from the general case):
Now solving for
q
1
yields
, the leader's optimal action:
This is the leader's best response to the reaction of the follower in equilibrium. The follower's actual can now be found by feeding this into its reaction function calculated earlier:
The Nash equilibria are all
. It is clear (if marginal costs are assumed to be zero - i.e. cost is essentially ignored) that the leader has a significant advantage. Intuitively, if the leader was no better off than the follower, it would simply adopt a Cournot competition strategy.
Plugging the follower's quantity, q2 back into the leader's best response function will not yield q1. This is because once leader has committed to an output and observed the followers it always wants to reduce its output ex-post. However its inability to do so is what allows it to receive higher profits than under cournot.
Economic analysis
An extensive-form representation is often used to analyze the Stackelberg leader-follower model. Also referred to as a “decision tree”, the model shows the combination of outputs and payoffs both firms have in the Stackelberg game
The image on the left depicts in extensive form a Stackelberg game. The payoffs are shown on the right. This example is fairly simple. There is a basic cost structure involving only marginal cost (there is no fixed cost). The demand function is linear and price elasticity of demand is 1. However, it illustrates the leader's advantage.
The follower wants to choose
q
2
to maximise its payoff
5000 −
q
1
−
q
2
−
c
2
. Taking the first order derivative and equating it to zero (for maximisation) yields
as the maximum value of
q
2
.
The leader wants to choose
q
1
to maximise its payoff
5000 −
q
1
−
q
2
−
c
1
. However, in equilibrium, it knows the follower will choose
q
2
as above. So in fact the leader wants to maximise its payoff
(by substituting
q
2
for the follower's best response function). By differentiation, the maximum payoff is given by
. Feeding this into the follower's best response function yields
. Suppose marginal costs were equal for the firms (so the leader has no market advantage other than first move) and in particular
c
1
=
c
2
= 1000
. The leader would produce 2000 and the follower would produce 1000. This would give the leader a profit (payoff) of two million and the follower a profit of one million. Simply by moving first, the leader has accrued twice the profit of the follower. However, Cournot profits here are 1.78 million apiece (strictly,
(16 / 9)10
6
apiece), so the leader has not gained much, but the follower has lost. However, this is example-specific. There may be cases where a Stackelberg leader has huge gains beyond Cournot profit that approach monopoly profits (for example, if the leader also had a large cost structure advantage, perhaps due to a better production function). There may also be cases where the follower actually enjoys higher profits than the leader, but only because it, say, has much lower costs.
Noncredible threats by the follower
If, after the leader had selected its equil