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

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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
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
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