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Consider a Robinson Crusoe's economy, where he has 10 hours to spend on gathering coconuts (L), or a leisure (R). That is, L + R = 10. He can produce coconuts based on the production. function given by C = 4√L, where C is the number of coconuts. He does not enjoy gathering coconuts, but does enjoy hist leisure time and eating coconuts. His utility function is given by u(C, R) = CR.

Consider a Robinson Crusoe's economy, where he has 10 hours to spend on gathering coconuts (L), or a leisure (R). That is, L + R = 10. He can produce coconuts based on the production. function given by C = 4√L, where C is the number of coconuts. He does not enjoy gathering coconuts, but does enjoy hist leisure time and eating coconuts. His utility function is given by u(C, R) = CR.

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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please answer part c,d,e here please

(2) Consider a Robinson Crusoe's economy, where he has 10 hours to spend on gathering coconuts (L), or a leisure (R). That is, L + R = 10. He can produce coconuts based on the production function given by C = 4√L, where C is the number of coconuts. He does not enjoy gathering coconuts, but does enjoy his leisure time and eating coconuts. His utility function is given by u(C, R) = CR. (a) Find the optimal allocation (C*, L*), in which Robinson Crusoe acts as a social planner. (b) Now consider a de-centralized economy. That is, think of this economy that consists of many individuals with the identical taste and technology. First, consider a firm's decision problem. The firm produces coconuts (C), employing workers (L). The firm hires L in a competitive market, given the wage rate (w). Without loss of generality, assume the price of coconuts to be Pc = 1. Given the market wage rate (w), how much labour would a firm hire in order to maximize its profits? Label your answer as Lp, and note that your answer must be a function of w. (c) Now consider a consumer. A consumer maximizes his/her utility (u= CR), has the endow- ment of II (think of a consumer being an owner of the firm), and also can work L hours at the wage rate (w). How many hours would a consumer work to maximize the utility, given the wage rate (w) and the endowment income (II)? label it as Lg, and note that your answer must be a function of w and II. (d) Find the equilibrium wage rate (w*). To find it out, you need to find what II is in terms of w. Then solve for w*. (e) Is the competitive equilibrium allocation Pareto optimal? Explain your answer.
Transcribed Image Text:(2) Consider a Robinson Crusoe's economy, where he has 10 hours to spend on gathering coconuts (L), or a leisure (R). That is, L + R = 10. He can produce coconuts based on the production function given by C = 4√L, where C is the number of coconuts. He does not enjoy gathering coconuts, but does enjoy his leisure time and eating coconuts. His utility function is given by u(C, R) = CR. (a) Find the optimal allocation (C*, L*), in which Robinson Crusoe acts as a social planner. (b) Now consider a de-centralized economy. That is, think of this economy that consists of many individuals with the identical taste and technology. First, consider a firm's decision problem. The firm produces coconuts (C), employing workers (L). The firm hires L in a competitive market, given the wage rate (w). Without loss of generality, assume the price of coconuts to be Pc = 1. Given the market wage rate (w), how much labour would a firm hire in order to maximize its profits? Label your answer as Lp, and note that your answer must be a function of w. (c) Now consider a consumer. A consumer maximizes his/her utility (u= CR), has the endow- ment of II (think of a consumer being an owner of the firm), and also can work L hours at the wage rate (w). How many hours would a consumer work to maximize the utility, given the wage rate (w) and the endowment income (II)? label it as Lg, and note that your answer must be a function of w and II. (d) Find the equilibrium wage rate (w*). To find it out, you need to find what II is in terms of w. Then solve for w*. (e) Is the competitive equilibrium allocation Pareto optimal? Explain your answer.
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