Cass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁20 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 20 to provide. When the price of the public good is p, 1's payoff is u₁ (1, 2) = a√√x₁+x2-pr₁ where a > 0 and 2's payoff is u2(x1, x2) = √√x1 + x2 − px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes. (b) Find all subgame perfect equilibria of this game for each (positive) value of a and p. Solution: Use backward induction. If 1 contributes 1, then 2's optimal action is to con- tribute z(x₁) = max x{₁,0}. Given this strategy for 2, 1's payoff to contributing 1 is - pai 2 a√₁-pr This payoff is maximized by choosing r1=0 if a < 2, x₁ = 0 or ₁ = 1/p² if a = 2, and x₁ = if a > 2. Therefore, the subgame perfect equilibria are as follows: i. If a < 2, there is a unique SPE given by x₁ = 0 and x2(x₁) = max {−₁,0}. ii. If a = 2, there are two SPE, one given by x₁ = 0 and x₂(x₁) = max {+ -21,0}, the other given by x₁ = 1/p² and x₂(₁): = max x{₁2-2²₁,0}. iii. If a > 2, there is a unique SPE given by x₁ = and x2(x₁) = max. x{₁,0}. (c) How does the total public good provision in part (b) compare to the Nash equilibrium provision when the consumers choose simultaneously? Solution: From class, total public good provision in the Nash equilibrium of the simul- taneous game is if a ≤ 1 and if a > 1. In the sequential game, total provision is OT rifa = 2, and if a > 2. Total provision is never higher in the and is strictly lower if 1 < a < 2. if a < 2, sequential game, if n ≤ otherwise. u(x₁, x₂(x₁)) =
Cass. Sup- 2. Consider the following sequential variant of the public goods game w pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁20 to provide of the public good. After observing 1's choice, 2 chooses a quantity 2 20 to provide. When the price of the public good is p, 1's payoff is u₁ (1, 2) = a√√x₁+x2-pr₁ where a > 0 and 2's payoff is u2(x1, x2) = √√x1 + x2 − px2. (a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes a positive amount. Solution: There are many such Nash equilibria. One is for 1 to contribute and for 2 to contribute 0 regardless of how much 1 contributes. (b) Find all subgame perfect equilibria of this game for each (positive) value of a and p. Solution: Use backward induction. If 1 contributes 1, then 2's optimal action is to con- tribute z(x₁) = max x{₁,0}. Given this strategy for 2, 1's payoff to contributing 1 is - pai 2 a√₁-pr This payoff is maximized by choosing r1=0 if a < 2, x₁ = 0 or ₁ = 1/p² if a = 2, and x₁ = if a > 2. Therefore, the subgame perfect equilibria are as follows: i. If a < 2, there is a unique SPE given by x₁ = 0 and x2(x₁) = max {−₁,0}. ii. If a = 2, there are two SPE, one given by x₁ = 0 and x₂(x₁) = max {+ -21,0}, the other given by x₁ = 1/p² and x₂(₁): = max x{₁2-2²₁,0}. iii. If a > 2, there is a unique SPE given by x₁ = and x2(x₁) = max. x{₁,0}. (c) How does the total public good provision in part (b) compare to the Nash equilibrium provision when the consumers choose simultaneously? Solution: From class, total public good provision in the Nash equilibrium of the simul- taneous game is if a ≤ 1 and if a > 1. In the sequential game, total provision is OT rifa = 2, and if a > 2. Total provision is never higher in the and is strictly lower if 1 < a < 2. if a < 2, sequential game, if n ≤ otherwise. u(x₁, x₂(x₁)) =
Chapter1: Making Economics Decisions
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Transcribed Image Text:Cass. Sup-
2. Consider the following sequential variant of the public goods game w
pose that there are 2 consumers, 1 and 2. First consumer 1 chooses a quantity ₁ ≥ 0 to
provide of the public good. After observing 1's choice, 2 chooses a quantity 2 ≥ 0 to provide.
When the price of the public good is p, 1's payoff is u₁(₁, 2) = a√x₁ + x₂-p²₁ where a > 0
and 2's payoff is u2(x1, x2) = √√T1+T2 — px2.
(a) Suppose that a = 1. Show that this game has a Nash equilibrium in which 1 contributes
a positive amount.
Solution: There are many such Nash equilibria. One is for 1 to contribute and for
2 to contribute 0 regardless of how much 1 contributes.
(b) Find all subgame perfect equilibria of this game for each (positive) value of a and p.
Solution: Use backward induction. If 1 contributes ₁, then 2's optimal action is to con-
tribute (1) max 7-2₁,0}. Given this strategy for 2, 1's payoff to contributing
₁ is
- px1
a√₁-pr
This payoff is maximized by choosing r1=0 if a < 2, x₁ = 0 or ₁ = 1/p² if a = 2, and
x₁ = if a > 2. Therefore, the subgame perfect equilibria are as follows:
if 1 ≤²
otherwise.
u(x₁, x₂(x₁)) =
i. If a < 2, there is a unique SPE given by x₁ = 0 and x2(x₁) = max.
{+ -21,0}.
ii. If a = 2, there are two SPE, one given by x₁ = 0 and x₂(x₁) = max {-₁,0},
{-1,0}.
I
the other given by x₁ = 1/p² and x₂(x₁) = max
iii. If a > 2, there is a unique SPE given by x₁ = and x₂(x₁) = max c{₁-2₁,0}.
(c) How does the total public good provision in part (b) compare to the Nash equilibrium
provision when the consumers choose simultaneously?
Solution: From class, total public good provision in the Nash equilibrium of the simul-
taneous game is
if a >1. In the sequential game, total provision is
if a > 2. Total provision is never higher in the
if a ≤ 1 and
ifa <2, or if a = 2, and
sequential game, and is strictly lower if 1 < a < 2.
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