First mover advantage in a Stackelberg game

Question

I am considering a simple game with two firms. Each firm faces the following demand function \begin{equation*} q_i(p_i,p_j)= a- b p_i + cp_j, \end{equation*} where $i,j\in \{1,2 \}$ and $i\neq j.$ Also, $b>c>0.$

Each firm sets its price to maximize its profit given as \begin{equation*} \Pi_i(p_i,p_j) = (p_i-\alpha_i) \times q_i(p_i,p_j), \end{equation*} where $\alpha_i$ is the production cost for firm $i$.

Assume Firm 1 is the Stackelberg leader and Firm 2 is the follower, that is, Firm 1 moves first and sets its price, then Firm 2 determines its price as a response to Firm 1's price.

I solve for the equilibrium prices by using backward induction and find a quite weird result. For example, when $a=500, b=25, c=20, \alpha_1=\alpha_2=10$, Firm 2's profit is higher than that of Firm 1.

This is an anomaly because Firm 1 has a first mover advantage yet lower profit. I suspect this is because of the demand function that I use. Can you help me with spotting the problem in this example?

Answer 1

In contrast to the quantity leadership model where one observe in the symmetric case a competition on the first move, i.e. leadership is preferred, the situation is reversed under a scenario of price leadership. In such a model with heterogeneous goods which are substitutes ($c>0$) -- of course, not total, we have some degree of product differentiation -- we get upward sloping reaction functions, which implies under the assumption of symmetry that following is preferred. However, as long as the reaction curves are downward sloping, leadership is also preferred in a price leadership oligopoly game.

The corresponding statement and proof can be found in Hal R. Varian, Microeconomic Analysis, Chap 16.7 Price Leadership, p. 300.

(Following preferred) If both firms have identical cost and demand functions and reaction curves are upward sloping, then each must prefer to be the follower to being the leader.

The leader has to reduce its output to support the price, whereas the follower can take the price as fixed and can produce whatever he wants, thus, firm 2 is able to make a higher profit than firm 1 in equilibrium (Varian, p. 300). To put it differently, in the symmetric case we observe a competition on the second move.

I am sorry, I had to know it, but I haven't it in my mind.

Answer 2

In price competition there is no first mover advantage. There is a second mover advantage instead. Simply stated the second mover observes the price of the first mover and sets a price $ε$ lower than the first movers price. Under the correct assumptions (elastic demand, responsive consumers etc.) he can make a higher profit than the first mover.

This is not the case in quantity competition. Cournot solved the simultaneous quantity competition game (substitute goods) and Stackelberg established that indeed, a firm that commits to a quantity prior to its competitor, gains a strategic advantage. In this model the market clears at a price determined by the total quantity that the two competitors sell. So, the first mover advantage comes (intuitively) from the fact that he manages to sell "more quantity". In contrast, in price competition, the second mover advantage comes from the fact that he can adjust optimally his price (not always, but under some general assumptions).