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.

ENGR.ECONOMIC ANALYSIS
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Chapter1: Making Economics Decisions
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Use second image for reference,

for part b here is referene;

The maximum profit is found at the tangency between the production function and the
isoprofit line.
In other words, the slope of the production function and the slope of the
isoprofit line must be the same. This is written as
MPL = w
 
where w is the slope of the isoprofit line. Then we get
sqrt1 / 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 production
function given by
D
C=√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.
I don't draw a diagram for this answer key. But I highly recommend that you go over with
the diagram in class.
The socially optimal choice is at the tangency between the production function and the
indifference curve in this model. That is,
MRS=MPL
Since we plot L on the horizontal axis and C on the vertical axis, the marginal rate of
substitution will be given by
MRS=
MUR
MUC
because R runs the opposite direction of L. So in this case, we get
MRS=
The slope of the production function is simply the marginal product of labour, which is
1
2√L
Thus the optimal choice will be at
MPL =
R
Expressing C and R in terms of L, we obtain
Solving this tangency condition, we get
с
R
1
VI
12-L 2√L
1
2√L
The corresponding C is then given by
-
√L x 2√L=12-L
2L=12 - L
3L=12
C* = √=√4=2
Transcribed Image Text:(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 production function given by D C=√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. I don't draw a diagram for this answer key. But I highly recommend that you go over with the diagram in class. The socially optimal choice is at the tangency between the production function and the indifference curve in this model. That is, MRS=MPL Since we plot L on the horizontal axis and C on the vertical axis, the marginal rate of substitution will be given by MRS= MUR MUC because R runs the opposite direction of L. So in this case, we get MRS= The slope of the production function is simply the marginal product of labour, which is 1 2√L Thus the optimal choice will be at MPL = R Expressing C and R in terms of L, we obtain Solving this tangency condition, we get с R 1 VI 12-L 2√L 1 2√L The corresponding C is then given by - √L x 2√L=12-L 2L=12 - L 3L=12 C* = √=√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 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 P = 1.
Given the market wage rate (w), how much labour would a firm hire in order to maximize
its profits? Label your answer as L, and note that your answer must be a function of w.
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 P = 1. Given the market wage rate (w), how much labour would a firm hire in order to maximize its profits? Label your answer as L, and note that your answer must be a function of w.
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