Stackelberg with 3 firms

Question

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.

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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 $q_3$

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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!

Answer 1

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.

Answer 2

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.