Stackelberg model with 3 symmetric firms

Question

Firm 1, the incumbent, sets quantity q1 first, then firm 2 enters the market and sets quantity q2, knowing q1. Lastly, firm 2 enters the market and sets quantity q3, knowing q1 and q2.

Firms are symmetric. There is a possibility of entry deterrence, however, for this case I assume that no firm will deter other firm’s entry.

I maximize profit of firm 3, taking q1 and q2 as given, get a reaction function q3 (q1, q2).

Then I maximize profit of firm 2, taking q1 as given but substituting q3 with reaction function of firm 3. I get a reaction function q2 (q1).

Then I maximize profit of firm 1, substituting quantities of both firms with their reaction functions. In a two-firm Stackelberg model I would get some number or a combination of constants, which is not dependent on the quantity of the “follower” firm. However, here I get a reaction function, which depends on q2.

I believe that I should’ve substituted reaction function q2 (q1) into q3 (q1, q2) to get a profit function of firm 1 which depends solely on q1. I’m unsure whether it is right from the theoretical/intuitive perspective. Firm 3 observes quantity set by firm 2 but does it observe its reaction function? Usually we assume that the “leader” correctly anticipates the quantity set by the “follower” but does it work the other way?

Answer 1

Suppose $q_3(q_1,q_2)$ denotes the reaction function of firm 3, and $q_2(q_1)$ denotes the reaction function of firm 2. To determine firm 1's choice, maximise the profit of firm 1 subject to the constraints that $q_3=q_3(q_1,q_2)$ and $q_2=q_2(q_1)$ i.e. if $P^d(Q)$ denotes the inverse demand and $C_1(q_1)$ is the cost function then optimal $q_1$ solves the following problem: \begin{eqnarray*} \max_{q_1} \ q_1P^d(q_1+q_2(q_1)+q_3(q_1,q_2(q_1)))-C_1(q_1)\end{eqnarray*} Observe that the solution is not a function of $q_2$ since the objective is just a function of $q_1$.