(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. (b) Determine the equilibrium conditions for the three markets using the resource constraints and the budget constraints. Provide a statement of the equilibrium. (e) Assume logarithmic utility U(e) - In(e) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Solve the social pla
(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. (b) Determine the equilibrium conditions for the three markets using the resource constraints and the budget constraints. Provide a statement of the equilibrium. (e) Assume logarithmic utility U(e) - In(e) and derive a closed form solution for consumption in both periods and savings for both types of agents. (d) Solve the social pla
Chapter3: Preferences And Utility
Section: Chapter Questions
Problem 3.9P
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![in the second, where
N = N1 + Ng.
Households have access to a credit market where they can borTow (s < 0) or save
s>0. The type 1 agent faces budget constraints
Y = c+s
rs!
where consumption for the type i agent in period j is denoted e. The type 2 agent
faces budget costraints
2 +rs
The resource constraints are
Nye+ NạG
Nic+ Nạ
Na2 =
(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.
(b) Determine the equilibrium conditions for the three markets using the resource
constraints and the budget constraints. Provide a statement of the equilibrium.
(c) Assume logarithmic utility U(e) = In(c) and derive a closed form solution for
consumption in both periods and savings for both types of agents.
(d) Solve the social planning problem. Compare the solution of the social planning
problem with the competitive equilibrium. Demonstrate that the decentralized
solution solves the social planning problem for a particular set of Pareto weights.
Explain how this is an example of the First Welfare Theorem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7e78add-7dac-4ed3-baa7-c3084220f262%2F2528b2e2-a8b6-490f-8dc7-66c0f3682c48%2Fx7lolgj_processed.png&w=3840&q=75)
Transcribed Image Text:in the second, where
N = N1 + Ng.
Households have access to a credit market where they can borTow (s < 0) or save
s>0. The type 1 agent faces budget constraints
Y = c+s
rs!
where consumption for the type i agent in period j is denoted e. The type 2 agent
faces budget costraints
2 +rs
The resource constraints are
Nye+ NạG
Nic+ Nạ
Na2 =
(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.
(b) Determine the equilibrium conditions for the three markets using the resource
constraints and the budget constraints. Provide a statement of the equilibrium.
(c) Assume logarithmic utility U(e) = In(c) and derive a closed form solution for
consumption in both periods and savings for both types of agents.
(d) Solve the social planning problem. Compare the solution of the social planning
problem with the competitive equilibrium. Demonstrate that the decentralized
solution solves the social planning problem for a particular set of Pareto weights.
Explain how this is an example of the First Welfare Theorem.
![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 Ñ 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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7e78add-7dac-4ed3-baa7-c3084220f262%2F2528b2e2-a8b6-490f-8dc7-66c0f3682c48%2F68zwd0g_processed.png&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 Ñ 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
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