A consumer has utility u(x₁, x2) = x₁ + x₁x2. Suppose that, because of a shortage of good 1, the government imposes a strict upper limit of ₁ on the quantity of good 1 that the consumer can consume. Assume throughout the following that w > P2. (a) Show that the consumer's preferences are strictly convex. Find the consumer's expenditure function. Solution: Using the answer to part (b), the indirect utility is v(p, w) = (1 + 2² W-P2 2P₁ (71 (1 + i)) P2 Using the identity v (p, e(p, u)) = u leads to e(p, u) = if ≤ 1, otherwise. 2√P1P2u - P2 if 1 <1, [P2 (-1) +P₁₁ otherwise. | Suppose that p₁ = P2, w = 3p₁, and F₁ = 1. If the government removes the restriction on consumption of good 1, the price of good 1 will rise from p₁ 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, >₁. Hence with the restriction, the consumer chooses x₁ = 1 and x2 = 2 giving utility 3. Without the restriction, the consumer chooses according to the interior solution we found in part (b), giving utility +2² (1+1-7-2² ) - 0²
A consumer has utility u(x₁, x2) = x₁ + x₁x2. Suppose that, because of a shortage of good 1, the government imposes a strict upper limit of ₁ on the quantity of good 1 that the consumer can consume. Assume throughout the following that w > P2. (a) Show that the consumer's preferences are strictly convex. Find the consumer's expenditure function. Solution: Using the answer to part (b), the indirect utility is v(p, w) = (1 + 2² W-P2 2P₁ (71 (1 + i)) P2 Using the identity v (p, e(p, u)) = u leads to e(p, u) = if ≤ 1, otherwise. 2√P1P2u - P2 if 1 <1, [P2 (-1) +P₁₁ otherwise. | Suppose that p₁ = P2, w = 3p₁, and F₁ = 1. If the government removes the restriction on consumption of good 1, the price of good 1 will rise from p₁ 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, >₁. Hence with the restriction, the consumer chooses x₁ = 1 and x2 = 2 giving utility 3. Without the restriction, the consumer chooses according to the interior solution we found in part (b), giving utility +2² (1+1-7-2² ) - 0²
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:A consumer has utility
u(x₁, x2) = x₁ + x1x2.
Suppose that, because of a shortage of good 1, the government imposes a strict upper limit
of ₁ on the quantity of good 1 that the consumer can consume. Assume throughout the
following that w>p2.
(a) Show that the consumer's preferences are strictly convex.
Find the consumer's expenditure function.
Solution: Using the answer to part (b), the indirect utility is
v(p, w) =
2p1 +
(1
(F₁(1+₁¹) otherwise.
P2
e(p, u) =
w+p2
Using the identity v (p, e(p, u)) = u leads to
W-P2
2p₂
if 1 <1,
2p1
w+P2 <1,
2√P1P2u P2
2p1
( P2 (-1) + P₁₁ otherwise.
if
| 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 p₁, and p2 and w will
remain the same. For what values of p₁ and p₁ does the consumer prefer that the limit
remain in place?
w+P₂
Solution: For the given values, >₁. Hence with the restriction, the consumer
chooses x₁ = 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
+ (1+²) -
|
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