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Exercise

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:

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Any help will be appreciated a lot. Thank you .

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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 : $$\exists q_1 \neq q_1^c, \ \pi_2(q_1, q_2^*) < \max_{q_2} \pi_2(q_1, q_2) $$

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