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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?

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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)

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    $\begingroup$ @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. $\endgroup$ – user315 May 11 '18 at 9:14
  • $\begingroup$ 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. $\endgroup$ – hllspwn May 11 '18 at 22:57
  • $\begingroup$ 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? $\endgroup$ – user315 May 12 '18 at 1:02
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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)

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  • $\begingroup$ Thank you for your answer. Okay it is just typo. Thank you, I corrected it. $\endgroup$ – user315 May 13 '18 at 0:45
  • $\begingroup$ 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? economics.stackexchange.com/questions/21964/… thanks a lot $\endgroup$ – user315 May 13 '18 at 0:50

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