<p>Let $\pi_1(q_1, q_2), \pi_2(q_1, q_2)$ denotes firm 1's profit and firm 2's profit respectively when firm 1 produces $q_1$ and firm 2 produces $q_2$.</p>

<p>In part (i), show that the given strategy profile is a Nash equilibrium by simply verifying the following :</p>

<ul>
<li>Firm 1 is playing it's best response to firm 2's strategy i.e.</li>
</ul>

<p>$$\pi_1(q_1^c, q_2^c) \geq \pi_1(q_1, q_2^*) \ \ \forall q_1 \neq q_1^c$$
where $q_2^* &gt; q_2^c$.</p>

<ul>
<li>Firm 2 is playing it's best response to firm 1's strategy i.e.</li>
</ul>

<p>$$\pi_2(q_1^c, q_2^c) \geq \pi_2(q_1^c, q_2) \ \ \forall q_2$$</p>

<p>In part (ii), you can show that the given strategy is not subgame perfect  by simply showing :
$$\exists q_1 \neq q_1^c, \ \pi_2(q_1, q_2^*) &lt; \max_{q_2} \pi_2(q_1, q_2) $$</p>

CC BY-SA 4.0
2018-06-13T13:39:53.350Z
Dear Amit. Thank you so  so so much. I have a regarding a past question that you have answered.If you want to have a look, I will be very pleased.   https://economics.stackexchange.com/questions/21902/optimization-problem-with-kuhn-tucker-conditions
Stefanos Makridis
CC BY-SA 4.0
2018-06-14T01:41:26.683Z
Dear Amit I have this question related to efficient allocation. Would it be possible to have look? Thank you in advance. https://economics.stackexchange.com/questions/22458/profit-maximization-and-allocative-efficiency-result
Stefanos Makridis
CC BY-SA 4.0
2018-06-13T12:47:30.553Z
2018-06-13T12:47:30.553Z
Amit
<p><a href="https://i.sstatic.net/tdkwv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tdkwv.png" alt="Exercise"></a></p>

<p>I am completely confused because I cannot find the leader's best response. I do not know if it is exactly the same as it was a Cournot game. What I did is:</p>

<p><a href="https://i.sstatic.net/pfuaN.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pfuaN.jpg" alt="Picture"></a></p>

<p>Any help will be appreciated a lot. Thank you .</p>

<microeconomics><game-theory><self-study>
Stackelberg setup
Stefanos Makridis
# Stackelberg setup

(Asked by: username_0 on 2018-06-12)

[![Exercise](https://i.sstatic.net/tdkwv.png)](https://i.sstatic.net/tdkwv.png)

I am completely confused because I cannot find the leader's best response. I do not know if it is exactly the same as it was a Cournot game. What I did is:

[![Picture](https://i.sstatic.net/pfuaN.jpg)](https://i.sstatic.net/pfuaN.jpg)

Any help will be appreciated a lot. Thank you .

**Tags:** microeconomics, game-theory, self-study

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## 1 answers

### Answer by username_1 on 2018-06-13. Score: 2 | Accepted answer

Let $\pi\_1(q\_1, q\_2), \pi\_2(q\_1, q\_2)$ denotes firm 1's profit and firm 2's profit respectively when firm 1 produces $q\_1$ and firm 2 produces $q\_2$.

In part (i), show that the given strategy profile is a Nash equilibrium by simply verifying the following :

* Firm 1 is playing it's best response to firm 2's strategy i.e.

$$\pi\_1(q\_1^c, q\_2^c) \geq \pi\_1(q\_1, q\_2^\*) \ \ \forall q\_1 \neq q\_1^c$$
where $q\_2^\* > q\_2^c$.

* Firm 2 is playing it's best response to firm 1's strategy i.e.

$$\pi\_2(q\_1^c, q\_2^c) \geq \pi\_2(q\_1^c, q\_2) \ \ \forall q\_2$$

In part (ii), you can show that the given strategy is not subgame perfect by simply showing :
$$\exists q\_1 \neq q\_1^c, \ \pi\_2(q\_1, q\_2^\*) < \max\_{q\_2} \pi\_2(q\_1, q\_2) $$

> **Comments:**
> **username_0**: Dear Amit. Thank you so  so so much. I have a regarding a past question that you have answered.If you want to have a look, I will be very pleased.   https://economics.stackexchange.com/questions/21902/optimization-problem-with-kuhn-tucker-conditions  
> **username_0**: Dear Amit I have this question related to efficient allocation. Would it be possible to have look? Thank you in advance. https://economics.stackexchange.com/questions/22458/profit-maximization-and-allocative-efficiency-result  

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