As usual, goods 1 and 2 are perfectly divisible, meaning that ₁ and 22 can take on any nonnegative real values and need not be integers. Good 3 is indivisible and unique, meaning that 23 must be either 0 or 1 (for example, good 3 could be an original painting). (a) Show that this consumer's preferences are monotone. Solution: The derivatives du/day and du/da2 are positive for x1, x2 > 0. Increasing 23 from 0 to 1 increases utility by 1/2. Therefore, if the quantity of all three goods increases, utility (strictly) increases. (b) Find this consumer's Marshallian demand. Solution: If x3 = 0, the optimal choice is (#₁, #2,83) = (25₁, 252,0). If x3 = 1, the optimal choice is (#1, #2, #3) = W 2p1 1 w - P3 2p2
As usual, goods 1 and 2 are perfectly divisible, meaning that ₁ and 22 can take on any nonnegative real values and need not be integers. Good 3 is indivisible and unique, meaning that 23 must be either 0 or 1 (for example, good 3 could be an original painting). (a) Show that this consumer's preferences are monotone. Solution: The derivatives du/day and du/da2 are positive for x1, x2 > 0. Increasing 23 from 0 to 1 increases utility by 1/2. Therefore, if the quantity of all three goods increases, utility (strictly) increases. (b) Find this consumer's Marshallian demand. Solution: If x3 = 0, the optimal choice is (#₁, #2,83) = (25₁, 252,0). If x3 = 1, the optimal choice is (#1, #2, #3) = W 2p1 1 w - P3 2p2
Chapter3: Preferences And Utility
Section: Chapter Questions
Problem 3.9P
Related questions
Question
please only do: if you can teach explain steps of how to solve each part use lengrange
![u(T1, T2, T3) = √√√1₂-2-³.
As usual, goods 1 and 2 are perfectly divisible, meaning that ₁ and 2 can take on any
nonnegative real values and need not be integers. Good 3 is indivisible and unique, meaning
that 23 must be either 0 or 1 (for example, good 3 could be an original painting).
(a) Show that this consumer's preferences are monotone.
Solution: The derivatives du/oxy and du/dr2 are positive for x1, x2 > 0. Increasing
13 from 0 to 1 increases utility by 1/2. Therefore, if the quantity of all three goods
increases, utility (strictly) increases.
(b) Find this consumer's Marshallian demand.
Solution: If x3 = 0, the optimal choice is
If x3 = 1, the optimal choice is
which simplifies to
W
(31,82,83) = (201₁, 2012, 0).
Therefore,
x(p, w):
Note that if p3>w, we must have x3 = 0. Otherwise, the optimal choice of x3 is 0 if
w-P3 w - P3
2p1 2p2
(F1, F2, F3) = (²
W W
2p12p2
12
w-P3 w-P3,
2p1 2p2
P3 ≥ √P1P2.
w-Pa
2p1 2pz
²,₁1).
(2P12P2,0)
1) if p3 ≤ w and p3 ≤ √P1P2,
if P3 w or P3 ≥ √P1P2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc93dd3a-e660-4464-bc8c-e382a2c34aae%2F6ff36a2b-8732-4033-a875-76dead455343%2Fldsnnq_processed.png&w=3840&q=75)
Transcribed Image Text:u(T1, T2, T3) = √√√1₂-2-³.
As usual, goods 1 and 2 are perfectly divisible, meaning that ₁ and 2 can take on any
nonnegative real values and need not be integers. Good 3 is indivisible and unique, meaning
that 23 must be either 0 or 1 (for example, good 3 could be an original painting).
(a) Show that this consumer's preferences are monotone.
Solution: The derivatives du/oxy and du/dr2 are positive for x1, x2 > 0. Increasing
13 from 0 to 1 increases utility by 1/2. Therefore, if the quantity of all three goods
increases, utility (strictly) increases.
(b) Find this consumer's Marshallian demand.
Solution: If x3 = 0, the optimal choice is
If x3 = 1, the optimal choice is
which simplifies to
W
(31,82,83) = (201₁, 2012, 0).
Therefore,
x(p, w):
Note that if p3>w, we must have x3 = 0. Otherwise, the optimal choice of x3 is 0 if
w-P3 w - P3
2p1 2p2
(F1, F2, F3) = (²
W W
2p12p2
12
w-P3 w-P3,
2p1 2p2
P3 ≥ √P1P2.
w-Pa
2p1 2pz
²,₁1).
(2P12P2,0)
1) if p3 ≤ w and p3 ≤ √P1P2,
if P3 w or P3 ≥ √P1P2
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 3 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you