u(x₁, x₂) = ₁ + 122. 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. Solution: Along any indifference curve x₁ + x₁x₂ = k, we have d²x₂ 1+x₂ = 2 da² > 0.
u(x₁, x₂) = ₁ + 122. 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. Solution: Along any indifference curve x₁ + x₁x₂ = k, we have d²x₂ 1+x₂ = 2 da² > 0.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:u(x₁, x₂) = x₁ + #172.
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.
Solution: Along any indifference curve x₁ + x₁x2 = k, we have
d²x2
dx²
1+22
#1
x2 =
= 2
Since preferences are monotone, this implies strict convexity.
(b) Find the consumer's Marshallian demand if the consumer cannot violate the government
limit.
Solution: By monotonicity, the budget constraint must bind. The FOCs for an interior
solution are
x(p, w) =
1 + x2 = Xp₁ and 1₁ = Xp₂.
Solving these together with the budget constraint gives
w-P2 and 21:
2p2
> 0.
Since w > p2 by assumption, these are both positive and the only constraint we need to
check is ₁ ≤₁. This leads to
w+p2 w-p2
2p1
2p2
(T1, w-piz
P2
w+p2
2p1
if 12 <1,
otherwise.
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