3. Consider an exchange environment with two consumers, Consumer 1 and Consumer 2, and two goods, a and b. Consumer 1 is initially endowed with e₁,a = 0 of good a and e₁,b = 10 of good b, while €1,6 Consumer 2 is initially endowed with e2,a = 10 of good a and €2,6 = 0 of good b. The utility of Consumer 1 from consuming x1, a ER+ of good a and 1,6 ER is u1(x1,a,x1,b) = x1,a + x1,b, while the utility of Consumer 2 from consuming x2,a Є R+ of good a and x2,6 ЄR is U2(x2,a, X2,b) = √√x2,a + x2,b. (a) Consider general equilibrium in the context of this environment in which the price of good b is normalized to 1. Compute the equilibrium price pt of good a and find the corresponding equilibrium quantities consumed by the various consumers. (b) Consider the game in which, simultaneously, Consumer 1 submits a “demand curve" and Consumer 2 submits a “supply curve” to a market mediator. In particular, the strategy space of Consumer 1 is [0,10], and the strategy d of Consumer 1 corresponds to the demand curve D₁ in which D₁(Pa) = d Pa is the amount of good a being demanded at price pa of good a. Similarly, the strategy space of Consumer 2 is [0,10]], and the strategy s of Consumer 2 corresponds to the supply curve S2 in which S2 (Pa) = −s + Pa is the amount of good a being supplied at price pa of good a. If Consumer 1 chooses a strategy d and Consumer 2 chooses a strategy s such that d≤s, then no trade occurs and each consumer consumes their initial endowments. If Consumer 1 chooses strategy d and Consumer 2 chooses a strategy s such that d>s, then trade occurs, the market mediator chooses the price pt at which D₁ (pa) = d − pa = −s + P₁ = p* S2(p), Consumer 1 consumes D₁(p) of good a and 10 - p* D₁ (pa) of good b, and Consumer 2 consumes 10 - - S2 (pt) of good a and pS2 (pt) of good b. Find a pure-strategy Nash equilibrium of this game in which trade occurs, and give the resulting equilibrium price p* of good a as well as the corresponding quantities of the goods consumed by the consumers. * *
3. Consider an exchange environment with two consumers, Consumer 1 and Consumer 2, and two goods, a and b. Consumer 1 is initially endowed with e₁,a = 0 of good a and e₁,b = 10 of good b, while €1,6 Consumer 2 is initially endowed with e2,a = 10 of good a and €2,6 = 0 of good b. The utility of Consumer 1 from consuming x1, a ER+ of good a and 1,6 ER is u1(x1,a,x1,b) = x1,a + x1,b, while the utility of Consumer 2 from consuming x2,a Є R+ of good a and x2,6 ЄR is U2(x2,a, X2,b) = √√x2,a + x2,b. (a) Consider general equilibrium in the context of this environment in which the price of good b is normalized to 1. Compute the equilibrium price pt of good a and find the corresponding equilibrium quantities consumed by the various consumers. (b) Consider the game in which, simultaneously, Consumer 1 submits a “demand curve" and Consumer 2 submits a “supply curve” to a market mediator. In particular, the strategy space of Consumer 1 is [0,10], and the strategy d of Consumer 1 corresponds to the demand curve D₁ in which D₁(Pa) = d Pa is the amount of good a being demanded at price pa of good a. Similarly, the strategy space of Consumer 2 is [0,10]], and the strategy s of Consumer 2 corresponds to the supply curve S2 in which S2 (Pa) = −s + Pa is the amount of good a being supplied at price pa of good a. If Consumer 1 chooses a strategy d and Consumer 2 chooses a strategy s such that d≤s, then no trade occurs and each consumer consumes their initial endowments. If Consumer 1 chooses strategy d and Consumer 2 chooses a strategy s such that d>s, then trade occurs, the market mediator chooses the price pt at which D₁ (pa) = d − pa = −s + P₁ = p* S2(p), Consumer 1 consumes D₁(p) of good a and 10 - p* D₁ (pa) of good b, and Consumer 2 consumes 10 - - S2 (pt) of good a and pS2 (pt) of good b. Find a pure-strategy Nash equilibrium of this game in which trade occurs, and give the resulting equilibrium price p* of good a as well as the corresponding quantities of the goods consumed by the consumers. * *
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
![3. Consider an exchange environment with two consumers, Consumer 1 and Consumer 2, and two goods,
a and b. Consumer 1 is initially endowed with e₁,a = 0 of good a and e₁,b = 10 of good b, while
€1,6
Consumer 2 is initially endowed with e2,a = 10 of good a and €2,6 = 0 of good b. The utility of
Consumer 1 from consuming x1, a ER+ of good a and 1,6 ER is
u1(x1,a,x1,b)
=
x1,a + x1,b,
while the utility of Consumer 2 from consuming x2,a Є R+ of good a and x2,6 ЄR is
U2(x2,a, X2,b)
=
√√x2,a + x2,b.
(a) Consider general equilibrium in the context of this environment in which the price of good b
is normalized to 1. Compute the equilibrium price pt of good a and find the corresponding
equilibrium quantities consumed by the various consumers.
(b) Consider the game in which, simultaneously, Consumer 1 submits a “demand curve" and Consumer
2 submits a “supply curve” to a market mediator. In particular, the strategy space of Consumer
1 is [0,10], and the strategy d of Consumer 1 corresponds to the demand curve D₁ in which
D₁(Pa) = d Pa is the amount of good a being demanded at price pa of good a. Similarly, the
strategy space of Consumer 2 is [0,10]], and the strategy s of Consumer 2 corresponds to the
supply curve S2 in which S2 (Pa) = −s + Pa is the amount of good a being supplied at price pa
of good a. If Consumer 1 chooses a strategy d and Consumer 2 chooses a strategy s such that
d≤s, then no trade occurs and each consumer consumes their initial endowments. If Consumer
1 chooses strategy d and Consumer 2 chooses a strategy s such that d>s, then trade occurs, the
market mediator chooses the price pt at which D₁ (pa) = d − pa = −s + P₁ =
p*
S2(p), Consumer 1
consumes D₁(p) of good a and 10 - p* D₁ (pa) of good b, and Consumer 2 consumes 10 - - S2 (pt)
of good a and pS2 (pt) of good b. Find a pure-strategy Nash equilibrium of this game in which
trade occurs, and give the resulting equilibrium price p* of good a as well as the corresponding
quantities of the goods consumed by the consumers.
*
*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33efa0ee-e3c4-4640-bf2a-d63d72536f00%2F0ed3df2c-8991-4861-952a-f7ee510ae314%2Ftdu2uvd_processed.png&w=3840&q=75)
Transcribed Image Text:3. Consider an exchange environment with two consumers, Consumer 1 and Consumer 2, and two goods,
a and b. Consumer 1 is initially endowed with e₁,a = 0 of good a and e₁,b = 10 of good b, while
€1,6
Consumer 2 is initially endowed with e2,a = 10 of good a and €2,6 = 0 of good b. The utility of
Consumer 1 from consuming x1, a ER+ of good a and 1,6 ER is
u1(x1,a,x1,b)
=
x1,a + x1,b,
while the utility of Consumer 2 from consuming x2,a Є R+ of good a and x2,6 ЄR is
U2(x2,a, X2,b)
=
√√x2,a + x2,b.
(a) Consider general equilibrium in the context of this environment in which the price of good b
is normalized to 1. Compute the equilibrium price pt of good a and find the corresponding
equilibrium quantities consumed by the various consumers.
(b) Consider the game in which, simultaneously, Consumer 1 submits a “demand curve" and Consumer
2 submits a “supply curve” to a market mediator. In particular, the strategy space of Consumer
1 is [0,10], and the strategy d of Consumer 1 corresponds to the demand curve D₁ in which
D₁(Pa) = d Pa is the amount of good a being demanded at price pa of good a. Similarly, the
strategy space of Consumer 2 is [0,10]], and the strategy s of Consumer 2 corresponds to the
supply curve S2 in which S2 (Pa) = −s + Pa is the amount of good a being supplied at price pa
of good a. If Consumer 1 chooses a strategy d and Consumer 2 chooses a strategy s such that
d≤s, then no trade occurs and each consumer consumes their initial endowments. If Consumer
1 chooses strategy d and Consumer 2 chooses a strategy s such that d>s, then trade occurs, the
market mediator chooses the price pt at which D₁ (pa) = d − pa = −s + P₁ =
p*
S2(p), Consumer 1
consumes D₁(p) of good a and 10 - p* D₁ (pa) of good b, and Consumer 2 consumes 10 - - S2 (pt)
of good a and pS2 (pt) of good b. Find a pure-strategy Nash equilibrium of this game in which
trade occurs, and give the resulting equilibrium price p* of good a as well as the corresponding
quantities of the goods consumed by the consumers.
*
*
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