Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+ which we defined, for any n≥0, as → R, Prove that u(x): -{ 21--1 1-n if ŋE [0,1) U (1,∞) In a if n = 1. 2--1 lim == Inz. →1 1-7 (7) Hint: Use L'Hôpital's Rule. Exercise 5. Consider the following two-period optimization problem we studied in class: max u (x1)+ẞu (x2), s.t. x1+px2=y, (1.22)ER² (8) where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0. 1. Find the bundle (1, 2) that solves (8). 2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con- sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.
Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+ which we defined, for any n≥0, as → R, Prove that u(x): -{ 21--1 1-n if ŋE [0,1) U (1,∞) In a if n = 1. 2--1 lim == Inz. →1 1-7 (7) Hint: Use L'Hôpital's Rule. Exercise 5. Consider the following two-period optimization problem we studied in class: max u (x1)+ẞu (x2), s.t. x1+px2=y, (1.22)ER² (8) where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0. 1. Find the bundle (1, 2) that solves (8). 2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con- sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
![Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+
which we defined, for any n≥0, as
→
R,
Prove that
u(x):
-{
21--1
1-n
if ŋE [0,1) U (1,∞)
In a
if n = 1.
2--1
lim
==
Inz.
→1
1-7
(7)
Hint: Use L'Hôpital's Rule.
Exercise 5. Consider the following two-period optimization problem we studied in class:
max u (x1)+ẞu (x2), s.t. x1+px2=y,
(1.22)ER²
(8)
where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount
factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income
expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0.
1. Find the bundle (1, 2) that solves (8).
2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con-
sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca420ce2-5907-4af8-acfd-f8810f9101e5%2Fdf5c03a7-1c71-4aea-8c5e-b86a8fe94442%2Fe1b2wv_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+
which we defined, for any n≥0, as
→
R,
Prove that
u(x):
-{
21--1
1-n
if ŋE [0,1) U (1,∞)
In a
if n = 1.
2--1
lim
==
Inz.
→1
1-7
(7)
Hint: Use L'Hôpital's Rule.
Exercise 5. Consider the following two-period optimization problem we studied in class:
max u (x1)+ẞu (x2), s.t. x1+px2=y,
(1.22)ER²
(8)
where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount
factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income
expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0.
1. Find the bundle (1, 2) that solves (8).
2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con-
sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.
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