<p>Suppose <span class="math-container">$q_3(q_1,q_2)$</span> denotes the reaction function of firm 3, and <span class="math-container">$q_2(q_1)$</span> denotes the reaction function of firm 2. To determine firm 1's choice, maximise the profit of firm 1 subject to the constraints that <span class="math-container">$q_3=q_3(q_1,q_2)$</span> and <span class="math-container">$q_2=q_2(q_1)$</span> i.e. if <span class="math-container">$P^d(Q)$</span> denotes the inverse demand and <span class="math-container">$C_1(q_1)$</span> is the cost function then optimal <span class="math-container">$q_1$</span> solves the following problem:
<span class="math-container">\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*}</span>
Observe that the solution is not a function of <span class="math-container">$q_2$</span> since the objective is just a function of <span class="math-container">$q_1$</span>.</p>

CC BY-SA 4.0
2023-12-12T19:31:29.200Z
2023-12-12T19:31:29.200Z
Amit
<p>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.</p>
<p>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.</p>
<p>I maximize profit of firm 3, taking q1 and q2 as given, get a reaction function q3 (q1, q2).</p>
<p>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).</p>
<p>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.</p>
<p>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?</p>

<microeconomics><profit-maximization><oligopoly>
Stackelberg model with 3 symmetric firms
Ksenia
# Stackelberg model with 3 symmetric firms

(Asked by: username_0 on 2023-12-12)

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?

**Tags:** microeconomics, profit-maximization, oligopoly

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## 1 answers

### Answer by username_1 on 2023-12-12. Score: 2

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 $q\_1$.


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