<p>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 <span class="math-container">$q_1$</span> as given. This will give you quantities <span class="math-container">$q_2$</span> and <span class="math-container">$q_3$</span> in terms of <span class="math-container">$q_1$</span> which you can then plug into the profit function of firm 1 and you can maximize <em>(i.e. find which <span class="math-container">$q_1$</span> firm 1 should choose to make sure the Nash equilibrium in stage 2 will be the most favourable Nash equilibrium possible for firm 1)</em>. So, step by step:</p>

<ol>
<li><p>Start with the second round, find the Nash equilibrium by solving the following two equations:
\
<span class="math-container">$$\pi_2'(q_1,q_2,q_3)=0\\
\pi_3'(q_1, q_2, q_3)=0$$</span>
Which will give you:
<span class="math-container">$$q_2=q_3=f(q_1)$$</span></p></li>
<li><p>plug this into the profit function of firm 1 and maximize this expression which has <span class="math-container">$q_1$</span> as choice variable.</p></li>
</ol>

CC BY-SA 4.0
2019-04-27T11:23:23.900Z
2019-05-13T09:56:49.570Z
2019-05-13T09:56:49.570Z
user18214
<p>In a scenario where there are no fixed or marginal costs, the leader gets <span class="math-container">$\frac{a}{2}$</span> of the market share, the next follower gets <span class="math-container">$\frac{a}{4}$</span>, third one gets <span class="math-container">$\frac{a}{8}$</span> and the <span class="math-container">$n^{th}$</span> firm gets <span class="math-container">$\frac{a}{2^{n}}$</span>. In the limiting case where the number of entrants tends to <span class="math-container">$\infty$</span>, the new entrants effectively become price takers.</p>

CC BY-SA 4.0
2019-05-02T21:00:59.647Z
2019-05-02T21:00:59.647Z
Sankalp Sharma
<p>I'm currently trying to solve the following problem:</p>

<blockquote>
  <p>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 <span class="math-container">$p(q) = A − Bq$</span>. The leader makes a production decision <span class="math-container">$q_1$</span>,
  then two followers make a simul- taneous decision about their
  production levels <span class="math-container">$q_2$</span> and <span class="math-container">$q_3$</span>. 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.</p>
</blockquote>

<hr>

<p><strong>My Workings</strong></p>

<p>I've tried to solve the problem using the following method:</p>

<p><strong>Leader (<span class="math-container">$q_1$</span>)</strong></p>

<p><span class="math-container">$$\begin{align} profit(q_1,q_2,q_3) &amp;= (A - B(q_1+q_2+q_3))q_1 -cq_1 -F \\
&amp; = Aq_1 - B(q_1+q_2+q_3)q_1 -cq_1 -F\\
&amp; = Aq_1 - Bq_1^2 - Bq_2q_1 - Bq_3q_1 - cq_1 -F\end{align}$$</span></p>

<p>The I took the derivative in regards to <span class="math-container">$q_1$</span> leaving me with this:
<span class="math-container">$$profit(q_1,q_2,q_3)= A - 2Bq_1 - Bq_2 -Bq_3 -c$$</span></p>

<p>Finally I just tried finding <span class="math-container">$q_1$</span>:</p>

<p><span class="math-container">$$q_1 = \frac{A-C-Bq_2-Bq_3}{2B}$$</span>
To put <span class="math-container">$q_1$</span> into the equation and solve it for <span class="math-container">$q_2$</span> and then <span class="math-container">$q_3$</span></p>

<hr>

<p><strong>The Problem:</strong></p>

<p>The begining of the solution in the answer sheet looks like this:</p>

<blockquote>
  <p>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</p>
</blockquote>

<p><span class="math-container">$$profit(q_i) = (A−B(q_1 +q_2 +q_3)−C)q_i ⇒ q_2 = q_3 = \frac{A − C}{3B} − \frac{q_1}{3}$$</span></p>

<p><strong>Question:</strong>
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 <span class="math-container">$3$</span> in <span class="math-container">$3B$</span> comes from? and why are we using <span class="math-container">$q_i$</span>? (also I don't quite understand why the method I used is incorrect)</p>

<p>Thank you very much for your help!</p>

<microeconomics><industrial-organisation><profit-maximization><firm><oligopoly>
Stackelberg with 3 firms
Fozoro
# Stackelberg with 3 firms

(Asked by: username_0 on 2019-04-27)

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!

**Tags:** microeconomics, industrial-organisation, profit-maximization, firm, oligopoly


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## 2 answers

### Answer by username_1 on 2019-04-27. Score: 2

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.


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### Answer by username_2 on 2019-05-02. Score: 0

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.


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