(a) State the maximization problem solved by each type of agent and derive the first- order and second-order conditions. Derive the solution using the implicit function theorem.
(a) State the maximization problem solved by each type of agent and derive the first- order and second-order conditions. Derive the solution using the implicit function theorem.
Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.14P
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![2. Constructing an Equilibrium
Households live two periods and have preferences
U(c) + BU(c2),
where 0 < B < 1 and U is the utility function and satisfies our usual assumptions.
There are N households in the economy. N\ of these households have endowment y
in the first period and no endowment in the second - these agents are called "Type 1".
The remaining N2 have no endowment in the first period and y2 in the second period
- these agents are called "Type 2." Hence the resources of the economy are
in the first period and
in the second, where
Ñ = N1 + N2.
Households have access to a credit market where they can borrow (s < 0) or save
s > 0. The type 1 agent faces budget constraints
Y1
c+s'
rs'
where consumption for the type i agent in period j is denoted c. The type 2 agent
faces budget constraints
G + s²
Y2 +rs? =
The resource constraints are
N1c + N2c
Nịc+ N2c
(a) State the maximization problem solved by each type of agent and derive the first-
order and second-order conditions. Derive the solution using the implicit function
theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7e78add-7dac-4ed3-baa7-c3084220f262%2F7616b0eb-44b4-4709-939e-89dde7ea6bb7%2F8mkxbl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Constructing an Equilibrium
Households live two periods and have preferences
U(c) + BU(c2),
where 0 < B < 1 and U is the utility function and satisfies our usual assumptions.
There are N households in the economy. N\ of these households have endowment y
in the first period and no endowment in the second - these agents are called "Type 1".
The remaining N2 have no endowment in the first period and y2 in the second period
- these agents are called "Type 2." Hence the resources of the economy are
in the first period and
in the second, where
Ñ = N1 + N2.
Households have access to a credit market where they can borrow (s < 0) or save
s > 0. The type 1 agent faces budget constraints
Y1
c+s'
rs'
where consumption for the type i agent in period j is denoted c. The type 2 agent
faces budget constraints
G + s²
Y2 +rs? =
The resource constraints are
N1c + N2c
Nịc+ N2c
(a) State the maximization problem solved by each type of agent and derive the first-
order and second-order conditions. Derive the solution using the implicit function
theorem.
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