Consider an economy with one consumption good, 100 identical consumers and 100 identical firms. Each consumer is endowed with one unit of time and one unit of capital. The agent can spend time either working or enjoying leisure. A represen- tative consumer's utility function is u (x, 1) = ln x + 2lnl, where x is consumption of goods and is leisure. Each firm hires consumers to work and rents capital from consumers to produce goods: y = ƒ (K,L) = K¹²L 1/2, where L is the amount of labor hired and K is the amount of capital rented. Both consumers and firms take goods price p, wage rate w and rental rate r as given. Normalize p = 1. (a) (b) (c) (d) (e) Set up a representative consumer's utility-maximization problem. Derive the Marshallian demands for consumption and leisure as functions of wage w and rental rate r. Set up a representative firm's profit-maximization problem. Write down the first-order conditions regarding the choices of K and L. Set up all the market-clearing conditions. Use the results from (a)-(c) to calculate equilibrium wage rate w*, rent rate r*, and the consumer's optimal choices of x* and 1*. Imagine a social planner who cares about all agents equally. The planner seeks to maximize a representative agent's utility subject to the feasi- bility constraint: max ln x+2ln s.t. 100x ≤ 100 (1 - 1) 1/2 . x,l Solve for the planner's solutions and compare them with the results in (d). Do the First and Second Welfare Theorems hold in this economy?
Consider an economy with one consumption good, 100 identical consumers and 100 identical firms. Each consumer is endowed with one unit of time and one unit of capital. The agent can spend time either working or enjoying leisure. A represen- tative consumer's utility function is u (x, 1) = ln x + 2lnl, where x is consumption of goods and is leisure. Each firm hires consumers to work and rents capital from consumers to produce goods: y = ƒ (K,L) = K¹²L 1/2, where L is the amount of labor hired and K is the amount of capital rented. Both consumers and firms take goods price p, wage rate w and rental rate r as given. Normalize p = 1. (a) (b) (c) (d) (e) Set up a representative consumer's utility-maximization problem. Derive the Marshallian demands for consumption and leisure as functions of wage w and rental rate r. Set up a representative firm's profit-maximization problem. Write down the first-order conditions regarding the choices of K and L. Set up all the market-clearing conditions. Use the results from (a)-(c) to calculate equilibrium wage rate w*, rent rate r*, and the consumer's optimal choices of x* and 1*. Imagine a social planner who cares about all agents equally. The planner seeks to maximize a representative agent's utility subject to the feasi- bility constraint: max ln x+2ln s.t. 100x ≤ 100 (1 - 1) 1/2 . x,l Solve for the planner's solutions and compare them with the results in (d). Do the First and Second Welfare Theorems hold in this economy?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Transcribed Image Text:Consider an economy with one consumption good, 100 identical consumers and 100
identical firms. Each consumer is endowed with one unit of time and one unit of
capital. The agent can spend time either working or enjoying leisure. A represen-
tative consumer's utility function is u (x, 1) = ln x + 2lnl, where x is consumption
of goods and is leisure. Each firm hires consumers to work and rents capital from
consumers to produce goods: y = ƒ (K,L) = K¹²L 1/2, where L is the amount of
labor hired and K is the amount of capital rented. Both consumers and firms take
goods price p, wage rate w and rental rate r as given. Normalize p = 1.
(a)
(b)
(c)
(d)
(e)
Set up a representative consumer's utility-maximization problem.
Derive the Marshallian demands for consumption and leisure as functions of
wage w and rental rate r.
Set up a representative firm's profit-maximization problem. Write
down the first-order conditions regarding the choices of K and L.
Set up all the market-clearing conditions.
Use the results from (a)-(c) to calculate equilibrium wage rate w*,
rent rate r*, and the consumer's optimal choices of x* and 1*.
Imagine a social planner who cares about all agents equally. The
planner seeks to maximize a representative agent's utility subject to the feasi-
bility constraint:
max ln x+2ln s.t. 100x ≤ 100 (1 - 1) 1/2 .
x,l
Solve for the planner's solutions and compare them with the results in (d). Do
the First and Second Welfare Theorems hold in this economy?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
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