7. Consider an economy where identical agents (of mass 1) live for two periods: youth (period 1) and old age (period 2). The utility function of a representative agent born at time t is given by u (C1t, C2+1) = log (C14) + B log (c2++1) where c denotes consumption in youth, c2 denotes consumption in old age, and 0 < B < 1 is the discount factor reflecting her time preference. In her youth the represen- tative agent supplies her endowment of 1 unit of labour inelastically and receives the market-determined wage rate wi. So in her youth the agent faces the budget constraint C14 + St = wo where są denotes her savings. When old, she just consumes her savings from youth plus the interest earning on her savings, sri+1; where r1+1 is the market- determined interest rate in period t + 1. That is, when old, her budget constraint is C21+1 = (1+r+1) s1 (a) Set up the agent's utility maximization problem by showing her choice variables clearly. (b) Write down the first order conditions for this maximization problem and derive the savings function. Explain how savings. st. if it does, depends on the interest rate T+1 The production function of the economy is given by Y, = AKÇ L}-ª 0 < a < 1, where K and L denote the amounts of capital and labour in the economy, respec- tively. Capital depreciates fully after use, that is. the rate of depreciation of capital is one. Factor markets being competitive, the equilibrium factor prices are given by their respective marginal products. (c) Derive the equilibrium wage rate (w.) of the economy in terms of K,.[ Keep in mind that the mass of agents is 1 and each agent supplies her endowment of 1 unit of labour inelastically.]
7. Consider an economy where identical agents (of mass 1) live for two periods: youth (period 1) and old age (period 2). The utility function of a representative agent born at time t is given by u (C1t, C2+1) = log (C14) + B log (c2++1) where c denotes consumption in youth, c2 denotes consumption in old age, and 0 < B < 1 is the discount factor reflecting her time preference. In her youth the represen- tative agent supplies her endowment of 1 unit of labour inelastically and receives the market-determined wage rate wi. So in her youth the agent faces the budget constraint C14 + St = wo where są denotes her savings. When old, she just consumes her savings from youth plus the interest earning on her savings, sri+1; where r1+1 is the market- determined interest rate in period t + 1. That is, when old, her budget constraint is C21+1 = (1+r+1) s1 (a) Set up the agent's utility maximization problem by showing her choice variables clearly. (b) Write down the first order conditions for this maximization problem and derive the savings function. Explain how savings. st. if it does, depends on the interest rate T+1 The production function of the economy is given by Y, = AKÇ L}-ª 0 < a < 1, where K and L denote the amounts of capital and labour in the economy, respec- tively. Capital depreciates fully after use, that is. the rate of depreciation of capital is one. Factor markets being competitive, the equilibrium factor prices are given by their respective marginal products. (c) Derive the equilibrium wage rate (w.) of the economy in terms of K,.[ Keep in mind that the mass of agents is 1 and each agent supplies her endowment of 1 unit of labour inelastically.]
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
100%
The question is mix of Micro and Macro. Please answer part a, b and c of the question. Please solve (or reject) it as soon as possible. Thank you.
![7. Consider an economy where identical agents (of mass 1 ) live for two periods: youth
(period 1) and old age (period 2). The utility function of a representative agent born at
time t is given by
u (C1t, C2,t+1) = log (C1,4) + B log (C2++1)
where ci denotes consumption in youth, c2 denotes consumption in old age, and 0 <
B < 1 is the discount factor reflecting her time preference. In her youth the represen-
tative agent supplies her endowment of 1 unit of labour inelastically and receives the
market-determined wage rate w1. So in her youth the agent faces the budget constraint
C1t + $t = wo where s; denotes her savings. When old, she just consumes her savings
from youth plus the interest earning on her savings, sri+1, where ri+1 is the market-
determined interest rate in period t + 1. That is, when old, her budget constraint is
C2,1+1 = (1+ r+1) s1
(a) Set up the agent's utility maximization problem by showing her choice variables
clearly.
(b) Write down the first order conditions for this maximization problem and derive the
savings function. Explain how savings. 8t. if it does, depends on the interest rate
Te+1
The production function of the economy is given by Y, = AK;L;¯ª 0 < a < 1,
where K and L denote the amounts of capital and labour in the economy, respec-
tively. Capital depreciates fully after use, that is. the rate of depreciation of capital
is one. Factor markets being competitive, the equilibrium factor prices are given by
their respective marginal products.
(c) Derive the equilibrium wage rate (w.) of the economy in terms of K,.[ Keep in mind
that the mass of agents is 1 and each agent supplies her endowment of 1 unit of
labour inelastically.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39f4412a-18c0-4f64-8bcf-4dffb42656da%2Fa1e88ca7-19d0-424f-8969-32eabc160f1b%2F3967wo_processed.png&w=3840&q=75)
Transcribed Image Text:7. Consider an economy where identical agents (of mass 1 ) live for two periods: youth
(period 1) and old age (period 2). The utility function of a representative agent born at
time t is given by
u (C1t, C2,t+1) = log (C1,4) + B log (C2++1)
where ci denotes consumption in youth, c2 denotes consumption in old age, and 0 <
B < 1 is the discount factor reflecting her time preference. In her youth the represen-
tative agent supplies her endowment of 1 unit of labour inelastically and receives the
market-determined wage rate w1. So in her youth the agent faces the budget constraint
C1t + $t = wo where s; denotes her savings. When old, she just consumes her savings
from youth plus the interest earning on her savings, sri+1, where ri+1 is the market-
determined interest rate in period t + 1. That is, when old, her budget constraint is
C2,1+1 = (1+ r+1) s1
(a) Set up the agent's utility maximization problem by showing her choice variables
clearly.
(b) Write down the first order conditions for this maximization problem and derive the
savings function. Explain how savings. 8t. if it does, depends on the interest rate
Te+1
The production function of the economy is given by Y, = AK;L;¯ª 0 < a < 1,
where K and L denote the amounts of capital and labour in the economy, respec-
tively. Capital depreciates fully after use, that is. the rate of depreciation of capital
is one. Factor markets being competitive, the equilibrium factor prices are given by
their respective marginal products.
(c) Derive the equilibrium wage rate (w.) of the economy in terms of K,.[ Keep in mind
that the mass of agents is 1 and each agent supplies her endowment of 1 unit of
labour inelastically.]
Transcribed Image Text:The role of the financial sector (banks, stock market, and so on) is to mobilize the
savings of households to bring it for effective use by the production sector. But the
financial sector does not work well and a fraction 0 < 0 < 1 of aggregate savings
gets lost (vanishes in thin air) in the process of intermediation.
(d) Derive the law of motion of capital (that is. express capital in period t + 1, Kt+1;
in terms of capital in period t, K;)
(e) Derive the steady state anount of capital of the economy.
(f) How does the steady state amount of capital depend on the inefficiency of the
financial sector 0?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education