There are two firms, whose production activity consumes some of the clean air that surrounds our planet. The total amount of clean air is K > 0, and any consumption of clean air comes out of this common resource. If firm i ∈ {1, 2} uses ki of clean air for its production, the remaining amount of clean air is K − k1 − k2. Each player derives utility from using ki for production and from the remainder of clean air. The payoff of firm i is given by ui(ki , kj ) = ln(ki) + ln(K − ki − kj ) j ≠ i ∈ {1, 2}. (a) Assuming that each firm chooses ki ∈ (0, K), to maximize its payoff function, derive the players’ best response functions and find a Nash equilibrium. (b) Is the equilibrium you found in (a) unique or not? What are equilibrium payoffs?
There are two firms, whose production activity consumes some of the clean air that surrounds our planet. The total amount of clean air is K > 0, and any consumption of clean air comes out of this common resource. If firm i ∈ {1, 2} uses ki of clean air for its production, the remaining amount of clean air is K − k1 − k2. Each player derives utility from using ki for production and from the remainder of clean air. The payoff of firm i is given by
ui(ki , kj ) = ln(ki) + ln(K − ki − kj ) j ≠ i ∈ {1, 2}.
(a) Assuming that each firm chooses ki ∈ (0, K), to maximize its payoff function, derive the players’ best response functions and find a Nash equilibrium.
(b) Is the equilibrium you found in (a) unique or not? What are equilibrium payoffs?
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