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 ; price is simply a function of total (industry) output, so is where the subscript 1 represents the leader and 2 represents the follower. Suppose firm i has the cost structure . 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 minus 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 that maximises given, i.e. given the output of the leader (firm 1), the output that maximises the follower's profit is found. Hence, the maximum of with respect to is to be found. First differentiate with respect to :
Setting this to zero for maximisation:
The values of 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, where is the follower's quantity as a function of the leader's quantity, namely the function calculated above. The best response is to find the value of that maximises given, 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 with respect to is to be found. First, differentiate with respect to :
Setting this to zero for maximisation:
Read more about this topic: Stackelberg Competition
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