Use second image for reference,for part b here is referene;The maximum profit is found at the tangency between the production function and theisoprofit line.In other words, the slope of the production function and the slope of theisoprofit line must be the same. This is written asMPL = w where w is the slope of the isoprofit line. Then we getsqrt1 / 2L = w =>1/2w = sqrtL=>L*D = 1/4w^2 (2) Consider a Robinson Crusoe's economy, where he has 12 hours to spend on gathering coconuts(L), or a leisure (R). That is, L+R=12. He can produce coconuts based on the productionfunction given byDC=√L,where C is the number of coconuts. He does not enjoy gathering coconuts, but does enjoy hisleisure time and eating coconuts. His utility function is given byu(C, R) = CR.(a) Find the optimal allocation (C", L"), in which Robinson Crusoe acts as a social planner.I don't draw a diagram for this answer key. But I highly recommend that you go over withthe diagram in class.The socially optimal choice is at the tangency between the production function and theindifference curve in this model. That is,MRS=MPLSince we plot L on the horizontal axis and C on the vertical axis, the marginal rate ofsubstitution will be given byMRS=MURMUCbecause R runs the opposite direction of L. So in this case, we getMRS=The slope of the production function is simply the marginal product of labour, which is12√LThus the optimal choice will be atMPL =RExpressing C and R in terms of L, we obtainSolving this tangency condition, we getсR1VI12-L 2√L12√LThe corresponding C is then given by-√L x 2√L=12-L2L=12 - L3L=12C* = √=√4=2 (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 productionfunction given byC = 4√L,where C is the number of coconuts. He does not enjoy gathering coconuts, but does enjoy hisleisure time and eating coconuts. His utility function is given byu(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 manyindividuals 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 competitivemarket, given the wage rate (w). Without loss of generality, assume the price of coconuts tobe P = 1.Given the market wage rate (w), how much labour would a firm hire in order to maximizeits profits? Label your answer as L, and note that your answer must be a function of w.

Given: Given: Production function: C=4L Utility function: 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 his 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 his leisure time and eating coconuts. His utility function is given by u(C, R) = CR.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.4P

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