<p>To answer your question I will assume that </p>

<p>$\bf{A1}$ firm 2 always "trust" in the output reported by firm 1, and</p>

<p>$\bf{A2}$ firm 1 always "keep its word" if announces an output.</p>

<p>Without $\bf{A1}$ and $\bf{A2}$ I think you need more information to solve the game.</p>

<p>Based on how the question is written, at the first period firm 1 has the opportunity to whether announce or not its output. So, lets say that firm 1 must to chose between $\{A, NA\}$ ($A$ for announce and $NA$ for not announce) at $t=1$. </p>

<p>Then, if firm 1 chose $A$ the resulting game will be the same as point i) of your question. In the same way, if firm 1 chose $NA$ the resulting game is the point ii) of your question. </p>

<p>Finally, you only need to obtain the utility of firm 1 for both cases to know what is the best action for firm 1 at $t=1$. The subgame perfect equilibrium must to include the action of firm 1 at $t=1$ and the quantities of both firms at $t=2$.</p>

<p>PS: In point ii) you have a mistake, the Cournot game is symmetric so there is impossible to get different quantities in a Nash equilibrium. (sorry for my bad english)</p>

CC BY-SA 4.0
2018-05-13T00:45:21.457Z
Thank you for your answer. Okay it is just typo. Thank you, I corrected it.
studentp
CC BY-SA 4.0
2018-05-13T00:50:07.143Z
I also have such a question. I did something but I am not sure. That is similar question to this one. Please also can you look at that? https://economics.stackexchange.com/questions/21964/question-on-oligopoly thanks a lot
studentp
CC BY-SA 4.0
2018-05-12T23:39:50.050Z
2018-05-12T23:46:13.763Z
2018-05-12T23:46:13.763Z
hllspwn
<p>There are two firs in the market. They produce perfect substitutes at cost $c(y_i)=y_i/3$ for i=1,2. The demand function is $p=1-(y_1+y_2)$ </p>

<p>Consider the Cournot competition where firms simultaneously produce their respective outputs. However firm 1 has the opportunity to announce the output it will produce to firm 2 before the firms have produced any output. How can I find the equilibrium quantities?</p>

<p>————————</p>

<p><strong>What I have done...</strong></p>

<p>For basic case, Cournot equilibrium</p>

<p>For firm 1, </p>

<p>$$max[(1-y_1-y_2)y_1-(y_1/3)]$$</p>

<p>FOCs</p>

<p>$$1-2y_1-y_2-(1/3)=0$$</p>

<p>$$y_1={2-3y_2\over 6}$$</p>

<p>For firm 2, </p>

<p>$$max[(1-y_1-y_2)y_2-(y_2/3)]$$</p>

<p>FOCs 
$$1-2y_2-y_1-(1/3)=0$$</p>

<p>$$y_2={2-3y_1\over 6}$$</p>

<p>So,</p>

<p>$$y_1=(1/3)-(1/2)(2-3y_1/6)$$
$$y^*_1=2/9$$
$$y_2^*=2/9$$</p>

<p>For Stackelberg eqn in basic cases </p>

<p>First firm is mover first 
$$max[(1-y_1-y_2)y_2-(y_2/3)]$$</p>

<p>FOCs 
$$1-2y_2-y_1-(1/3)=0$$</p>

<p>$$y_2={2-3y_1\over 6}$$</p>

<p>For firm 1,</p>

<p>$$max[(1-y_1-y_2)y_1-(y_1/3)]$$</p>

<p>$$max[(1-y_1-({2-3y_1\over 6}))y_1-(y_1/3)]$$</p>

<p>FOCs</p>

<p>$$1-2y_1-(1/3)y_1-y_1-(1/3)=0$$</p>

<p>$$y_1^*=1/5$$</p>

<p>$$y_2^*=7/30$$</p>

<p>I just find only Stackelberg equilibrium and Cournot equilibrium in basic cases. </p>

<p>But I cannot find the part that I write above. How can I solve this part? </p>

<p>Thank you. </p>

<hr>

<p>Edit: (I just post the original version of my question)</p>

<p><a href="https://i.sstatic.net/OSbFP.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OSbFP.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2018-05-11T09:14:02.680Z
@Aneconomist my every question is totally different. And typing is not problem. I edited. And these are exactly not homework. Also I always add my solution.
studentp
CC BY-SA 4.0
2018-05-11T22:57:45.913Z
Could you be more specific with the structure of the game? Is this an extensive game? when you said "firm 1 has the opportunity to announce", means that actions set for firm 1 is in {announce, not announce} at the first stage or not? That details are very important for the resolution of a game.
hllspwn
CC BY-SA 4.0
2018-05-12T01:02:26.510Z
Dear @hllspwn no information that you said. I also add the original version of may question. I asked Part(iii). What is your idea in order to solve it?
studentp
<microeconomics><game-theory><self-study><competitive-equilibrium><profit-maximization>
Cournot equilibrium question
studentp
# Cournot equilibrium question

(Asked by: username_0 on 2018-05-11)

There are two firs in the market. They produce perfect substitutes at cost $c(y\_i)=y\_i/3$ for i=1,2. The demand function is $p=1-(y\_1+y\_2)$

Consider the Cournot competition where firms simultaneously produce their respective outputs. However firm 1 has the opportunity to announce the output it will produce to firm 2 before the firms have produced any output. How can I find the equilibrium quantities?

————————

**What I have done...**

For basic case, Cournot equilibrium

For firm 1,

$$max[(1-y\_1-y\_2)y\_1-(y\_1/3)]$$

FOCs

$$1-2y\_1-y\_2-(1/3)=0$$

$$y\_1={2-3y\_2\over 6}$$

For firm 2,

$$max[(1-y\_1-y\_2)y\_2-(y\_2/3)]$$

FOCs
$$1-2y\_2-y\_1-(1/3)=0$$

$$y\_2={2-3y\_1\over 6}$$

So,

$$y\_1=(1/3)-(1/2)(2-3y\_1/6)$$
$$y^\*\_1=2/9$$
$$y\_2^\*=2/9$$

For Stackelberg eqn in basic cases

First firm is mover first
$$max[(1-y\_1-y\_2)y\_2-(y\_2/3)]$$

FOCs
$$1-2y\_2-y\_1-(1/3)=0$$

$$y\_2={2-3y\_1\over 6}$$

For firm 1,

$$max[(1-y\_1-y\_2)y\_1-(y\_1/3)]$$

$$max[(1-y\_1-({2-3y\_1\over 6}))y\_1-(y\_1/3)]$$

FOCs

$$1-2y\_1-(1/3)y\_1-y\_1-(1/3)=0$$

$$y\_1^\*=1/5$$

$$y\_2^\*=7/30$$

I just find only Stackelberg equilibrium and Cournot equilibrium in basic cases.

But I cannot find the part that I write above. How can I solve this part?

Thank you.

---

Edit: (I just post the original version of my question)

[![enter image description here](https://i.sstatic.net/OSbFP.jpg)](https://i.sstatic.net/OSbFP.jpg)

**Tags:** microeconomics, game-theory, self-study, competitive-equilibrium, profit-maximization
> **Comments:**
> **username_0**: @Aneconomist my every question is totally different. And typing is not problem. I edited. And these are exactly not homework. Also I always add my solution.  
> **username_1**: Could you be more specific with the structure of the game? Is this an extensive game? when you said "firm 1 has the opportunity to announce", means that actions set for firm 1 is in {announce, not announce} at the first stage or not? That details are very important for the resolution of a game.  
> **username_0**: Dear @hllspwn no information that you said. I also add the original version of may question. I asked Part(iii). What is your idea in order to solve it?  

---

## 1 answers

### Answer by username_1 on 2018-05-12. Score: 1 | Accepted answer

To answer your question I will assume that

$\bf{A1}$ firm 2 always "trust" in the output reported by firm 1, and

$\bf{A2}$ firm 1 always "keep its word" if announces an output.

Without $\bf{A1}$ and $\bf{A2}$ I think you need more information to solve the game.

Based on how the question is written, at the first period firm 1 has the opportunity to whether announce or not its output. So, lets say that firm 1 must to chose between $\{A, NA\}$ ($A$ for announce and $NA$ for not announce) at $t=1$.

Then, if firm 1 chose $A$ the resulting game will be the same as point i) of your question. In the same way, if firm 1 chose $NA$ the resulting game is the point ii) of your question.

Finally, you only need to obtain the utility of firm 1 for both cases to know what is the best action for firm 1 at $t=1$. The subgame perfect equilibrium must to include the action of firm 1 at $t=1$ and the quantities of both firms at $t=2$.

PS: In point ii) you have a mistake, the Cournot game is symmetric so there is impossible to get different quantities in a Nash equilibrium. (sorry for my bad english)

> **Comments:**
> **username_0**: Thank you for your answer. Okay it is just typo. Thank you, I corrected it.  
> **username_0**: I also have such a question. I did something but I am not sure. That is similar question to this one. Please also can you look at that? https://economics.stackexchange.com/questions/21964/question-on-oligopoly thanks a lot  

---

