Exercises 06 -- solutions(1)

Exercises 06 -- solutions(1) - ECO 444 Exercises...

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ECO 444 – Exercises 6 (Solutions) Konrad Grabiszewski Exercise 1 (a) We begin with Firm B (since they are the last to move). Firm B chooses p B to maximize their utility function. u B ( p A , p B ) = ( a - p B + p A ) p B (1) FOD and SOD du B ( p A , p B ) dp B = a - 2 p B + p A (2) d 2 u B ( p A , p B ) dp 2 B = - 2 < 0 (3) We derive the best-response function of Firm B, ˜ p B . ˜ p B = 1 2 ( a + p A ) (4) Now, we turn to Firm A that chooses p A to maximize their utility function. u A ( p A , ˜ p B ) = ( a - p A + ˜ p B ) p A = ( a - p A + 1 2 ( a + p A )) p A = 1 . 5 ap A - 0 . 5 p 2 A (5) FOD and SOD du A ( p A , ˜ p B ) dp B = 1 . 5 a - p A (6) d 2 u A ( p A , ˜ p B ) dp 2 B = - 1 < 0 (7) From FOC we obtain the equilibrium price of Firm A. p * A = 1 . 5 a (8) 1
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Next, we obtain the equilibrium price of Firm B. p * B = 1 . 25 a (9) Finally, we derive utilities at equilibrium. u A ( p * A , p * B ) = 1 . 125 a 2 (10) u B ( p * A , p * B ) = 1 . 5625 a 2 (11) (b) In game (i), Firm A gains 1 . 125 a 2 . In game (ii), Firm A gains 1 . 5625 a 2 . Hence, game (ii) is the one that I would recommend to Firm A. Exercise 2 (a) We begin with Firm B (since they are the last to move). Firm B chooses s B
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