2
$\begingroup$

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!

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.