(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.

Microeconomic Theory
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ISBN:9781337517942
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Chapter4: Utility Maximization And Choice
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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.
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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