<p>You are right that you first have to find F3's best-response function. F1 and F2 take as given this reaction of F3 to whatever they produce.
Hence, you plug this best-response function into the incumbents' profit maximization problem. In that way, you take care of the fact that the incumbents anticipate F3's reaction, indirectly determining <span class="math-container">$q_3$</span>.</p>

<p>You should also check if the resulting reaction quantity <span class="math-container">$q_3$</span> is non-negative! It can be optimal to deter the entry of F3 completely.</p>

CC BY-SA 4.0
2019-05-22T23:59:33.137Z
I tried this, but i just get: Max: (16-q1-(6-0.5q1-0.5q2)q1-4q1-->         
0.5q3q1-0.5q2q1 = 0 -->                                                                                             FOC: -0.5q3-0.5q2 = 0   -->                                                                                     How do i determine the actual value of q3?
Oliver
CC BY-SA 4.0
2019-05-23T08:33:09.547Z
The reaction function of F3 is some $q_3(q_1,q_2)$. Plugged into the maximization problem of F1, you get some $q_1(q_2)$. Analoguously, you can find $q_2(q_1)$. Plug them into each other for some $q_1^*$ and $q_2^*$, which then solve $q_3^*=q_3(q_1^*,q_2^*)$.
Bayesian
CC BY-SA 4.0
2019-05-22T10:33:52.513Z
2019-05-22T10:33:52.513Z
Bayesian
<p>Imagine that there are 3 firms in a monopolistic market, F1, F2 and F3. Firms 1 and 2 are incumbent firms and act simultaneously whereas Firm 3 observes the actions of both firms before deciding whether to enter. </p>

<p>All firms face a market price of <span class="math-container">$P(Q) = 16-q_1-q_2-q_3$</span>. Incumbent firms face a production cost of <span class="math-container">$C_i (q_i) = 4q_i$</span> for <span class="math-container">$i= 1,2$</span>. Firm 3 faces a production cost of <span class="math-container">$C_3(q_3) = 4 + 4q_3$</span>. Find both firm 3 and firm 1's payoff.</p>

<p>I understand that I need to take firm 3's best response function which is 
<span class="math-container">$$0.5  (12-q_2-q_1)$$</span> 
but I am not sure where to go from there. Do I substitute this back into the price equation? I tried doing this but it wouldn't yield any definitive answers. Thanks!</p>

CC BY-SA 4.0
2019-05-22T10:53:49.703Z
I just posted a hint because we generally don't do homework questions here.
Bayesian
CC BY-SA 4.0
2019-05-23T05:21:00.003Z
Your best response is not right. you should have $q_3$ as a function of $q_1$ and $q_2$. I'll just give you the road map. Player 3 observes $q_1$ and $q_2$ so take these two as parameters. Then, his profit function is $\pi(q_3)=(16-q_1-q_2-q_3)*q_3-(4+4q_3)$. To obtain the best response you should take the derivative $\pi'(q_3)$ and set it equal to zero. You solve that equation for $q_3$ and that is F3's best response. You plug your result in F1's and F2's profit function. Lastly, you take FOC's of these two and solve for $q_1^*, q_2^*$, and plug these into F3's best response to obtain $q_3^*$
Regio
CC BY-SA 4.0
2019-05-25T02:49:37.737Z
Sorry i miss-wrote the function I have fixed it now, thanks
Oliver
<microeconomics><game-theory><industrial-organisation><oligopoly>
Stackelberg Oligopoly 3 firms
Oliver
# Stackelberg Oligopoly 3 firms

(Asked by: username_0 on 2019-05-22)

Imagine that there are 3 firms in a monopolistic market, F1, F2 and F3. Firms 1 and 2 are incumbent firms and act simultaneously whereas Firm 3 observes the actions of both firms before deciding whether to enter.

All firms face a market price of $P(Q) = 16-q\_1-q\_2-q\_3$. Incumbent firms face a production cost of $C\_i (q\_i) = 4q\_i$ for $i= 1,2$. Firm 3 faces a production cost of $C\_3(q\_3) = 4 + 4q\_3$. Find both firm 3 and firm 1's payoff.

I understand that I need to take firm 3's best response function which is
$$0.5 (12-q\_2-q\_1)$$
but I am not sure where to go from there. Do I substitute this back into the price equation? I tried doing this but it wouldn't yield any definitive answers. Thanks!

**Tags:** microeconomics, game-theory, industrial-organisation, oligopoly
> **Comments:**
> **username_1**: I just posted a hint because we generally don't do homework questions here.  
> **username_2**: Your best response is not right. you should have $q_3$ as a function of $q_1$ and $q_2$. I'll just give you the road map. Player 3 observes $q_1$ and $q_2$ so take these two as parameters. Then, his profit function is $\pi(q_3)=(16-q_1-q_2-q_3)*q_3-(4+4q_3)$. To obtain the best response you should take the derivative $\pi'(q_3)$ and set it equal to zero. You solve that equation for $q_3$ and that is F3's best response. You plug your result in F1's and F2's profit function. Lastly, you take FOC's of these two and solve for $q_1^*, q_2^*$, and plug these into F3's best response to obtain $q_3^*$  
> **username_0**: Sorry i miss-wrote the function I have fixed it now, thanks  

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## 1 answers

### Answer by username_1 on 2019-05-22. Score: 1

You are right that you first have to find F3's best-response function. F1 and F2 take as given this reaction of F3 to whatever they produce.
Hence, you plug this best-response function into the incumbents' profit maximization problem. In that way, you take care of the fact that the incumbents anticipate F3's reaction, indirectly determining $q\_3$.

You should also check if the resulting reaction quantity $q\_3$ is non-negative! It can be optimal to deter the entry of F3 completely.

> **Comments:**
> **username_0**: I tried this, but i just get: Max: (16-q1-(6-0.5q1-0.5q2)q1-4q1-->          0.5q3q1-0.5q2q1 = 0 -->                                                                                             FOC: -0.5q3-0.5q2 = 0   -->                                                                                     How do i determine the actual value of q3?  
> **username_1**: The reaction function of F3 is some $q_3(q_1,q_2)$. Plugged into the maximization problem of F1, you get some $q_1(q_2)$. Analoguously, you can find $q_2(q_1)$. Plug them into each other for some $q_1^*$ and $q_2^*$, which then solve $q_3^*=q_3(q_1^*,q_2^*)$.  

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