A consumer has utility (1,₂)=1+12. Suppose that, because of a shortage of good 1, the government imposes a strict upper limit of 7₁ on the quantity of good 1 that the consumer can consume. Assume throughout the following that w>p2. Suppose that p₁ = P2, w = 3p₁, and 7₁ = 1. If the government removes the restriction on consumption of good 1, the price of good 1 will rise from pi to p₁, and p2 and we will remain the same. For what values of p₁ and p does the consumer prefer that the limit remain in place? Solution: For the given values, >7₁. Hence with the restriction, the consumer chooses ₁ = 1 and 2 = 2 giving utility 3. Without the restriction, the consumer chooses according to the interior solution we found in part (b), giving utility x(p, w) = v(p, w) = w+P2 2+² (1+ 2p/ w+p2 w-P2 2p1 2pa F11 e(p, u) = The interior solution is indeed a max because of converity and monotonicity. Find the consumer's expenditure function. Solution: Using the answer to part (b), the indirect utility is w+p₂ 2p1 P2 4p₁ 2p2 w+p₂ if 1, 2p1 -Z₁ otherwise. P2 Using the identity v (p, e(p, u)) = u leads to 10+2 531, ₁(1+²) otherwise. P2 M-P2 2p2 w+P2 if 1, 2p1 [2√Pip2u - P2 [P2 (-1) +P₁1 otherwise.
A consumer has utility (1,₂)=1+12. Suppose that, because of a shortage of good 1, the government imposes a strict upper limit of 7₁ on the quantity of good 1 that the consumer can consume. Assume throughout the following that w>p2. Suppose that p₁ = P2, w = 3p₁, and 7₁ = 1. If the government removes the restriction on consumption of good 1, the price of good 1 will rise from pi to p₁, and p2 and we will remain the same. For what values of p₁ and p does the consumer prefer that the limit remain in place? Solution: For the given values, >7₁. Hence with the restriction, the consumer chooses ₁ = 1 and 2 = 2 giving utility 3. Without the restriction, the consumer chooses according to the interior solution we found in part (b), giving utility x(p, w) = v(p, w) = w+P2 2+² (1+ 2p/ w+p2 w-P2 2p1 2pa F11 e(p, u) = The interior solution is indeed a max because of converity and monotonicity. Find the consumer's expenditure function. Solution: Using the answer to part (b), the indirect utility is w+p₂ 2p1 P2 4p₁ 2p2 w+p₂ if 1, 2p1 -Z₁ otherwise. P2 Using the identity v (p, e(p, u)) = u leads to 10+2 531, ₁(1+²) otherwise. P2 M-P2 2p2 w+P2 if 1, 2p1 [2√Pip2u - P2 [P2 (-1) +P₁1 otherwise.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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please only do: if you can teach explain steps of how to solve each part how was formula retrieve?
Suppose that p1 = p2, w = 3p1, and x1 = 1. If the government removes the restriction
on consumption of good 1, the
1, and p2 and w will
remain the same. For what values of p1 and p0
1 does the consumer prefer that the limit
remain in place?
Transcribed Image Text:A consumer has utility
u(x1, x₂) = x₂ + x1x₂.
Suppose that, because of a shortage of good 1, the government imposes a strict upper limit
of on the quantity of good 1 that the consumer can consume. Assume throughout the
following that w>p2.
Suppose that p₁ = P2, w = 3p₁, and ₁= 1. If the government removes the restriction
on consumption of good 1, the price of good 1 will rise from pi to ph, and p2 and we will
remain the same. For what values of p₁ and p does the consumer prefer that the limit
remain in place?
Solution: For the given values, >₁. Hence with the restriction, the consumer
chooses ₁ = 1 and 2 = 2 giving utility 3. Without the restriction, the consumer chooses
according to the interior solution we found in part (b), giving utility
x(p, w) =
v(p, w) =
w + P2
27/12 (1+
2p
w+P2 W-P2
2p1 2p2
F1,
e(p, u) =
w-p
P2
w+p₂ (1+ -P2
2p1
2p2
The interior solution is indeed a max because of convexity and monotonicity.
Find the consumer's expenditure function.
Solution: Using the answer to part (b), the indirect utility is
Using the identity v (p, e(p, u)) = u leads to
w-P2
2p2
( 7₁ (1 + 1-P1²₁)
P2
=
2) if!
4p1
Pi
if 10² ≤ 1,
otherwise.
w+p₂
2p1
otherwise.
≤1,
2√P₁P2u - p2
w+P2
if SII,
2p1
(P₂ (-1) + P₁₁ otherwise.
P2
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