# Stackelberg with 3 firms

I'm currently trying to solve the following problem:

Stackelberg with 3 firms Imagine there are three firms on a monopolistically competitive market. The marginal cost of produc- tion in each firm is c. The demand is $$p(q) = A − Bq$$. The leader makes a production decision $$q_1$$, then two followers make a simul- taneous decision about their production levels $$q_2$$ and $$q_3$$. Calculate the quantity produced by firms in this economy, and compare it to Cournot outcome with 3 firms and to Stackelberg outcome with 2 firms.

My Workings

I've tried to solve the problem using the following method:

Leader ($$q_1$$)

\begin{align} profit(q_1,q_2,q_3) &= (A - B(q_1+q_2+q_3))q_1 -cq_1 -F \\ & = Aq_1 - B(q_1+q_2+q_3)q_1 -cq_1 -F\\ & = Aq_1 - Bq_1^2 - Bq_2q_1 - Bq_3q_1 - cq_1 -F\end{align}

The I took the derivative in regards to $$q_1$$ leaving me with this: $$profit(q_1,q_2,q_3)= A - 2Bq_1 - Bq_2 -Bq_3 -c$$

Finally I just tried finding $$q_1$$:

$$q_1 = \frac{A-C-Bq_2-Bq_3}{2B}$$ To put $$q_1$$ into the equation and solve it for $$q_2$$ and then $$q_3$$

The Problem:

The begining of the solution in the answer sheet looks like this:

Solution For Stackelberg with two followers, after firm 1 made its move, agents 2 and 3 are making their move simultaneously knowing q1. So, both firms 2 and 3 maximize

$$profit(q_i) = (A−B(q_1 +q_2 +q_3)−C)q_i ⇒ q_2 = q_3 = \frac{A − C}{3B} − \frac{q_1}{3}$$

Question: as you can see my workings look nowhere near the answer, I've tried solving the problem using there method but I don't really understand were the $$3$$ in $$3B$$ comes from? and why are we using $$q_i$$? (also I don't quite understand why the method I used is incorrect)

Thank you very much for your help!

Start with the second stage, this is just Cournot competition between firm 2 and firm 3. You can solve this for the Nash equilibrium by setting the first order condition for firm 2 and firm 3 and solving these two equations, taking $$q_1$$ as given. This will give you quantities $$q_2$$ and $$q_3$$ in terms of $$q_1$$ which you can then plug into the profit function of firm 1 and you can maximize (i.e. find which $$q_1$$ firm 1 should choose to make sure the Nash equilibrium in stage 2 will be the most favourable Nash equilibrium possible for firm 1). So, step by step:
1. Start with the second round, find the Nash equilibrium by solving the following two equations: \ $$\pi_2'(q_1,q_2,q_3)=0\\ \pi_3'(q_1, q_2, q_3)=0$$ Which will give you: $$q_2=q_3=f(q_1)$$
2. plug this into the profit function of firm 1 and maximize this expression which has $$q_1$$ as choice variable.
In a scenario where there are no fixed or marginal costs, the leader gets $$\frac{a}{2}$$ of the market share, the next follower gets $$\frac{a}{4}$$, third one gets $$\frac{a}{8}$$ and the $$n^{th}$$ firm gets $$\frac{a}{2^{n}}$$. In the limiting case where the number of entrants tends to $$\infty$$, the new entrants effectively become price takers.