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. Since preferences are monotone, this implies strict convexity. (b) Find the consumer's Marshallian demand if the consumer cannot violate the government
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. Since preferences are monotone, this implies strict convexity. (b) Find the consumer's Marshallian demand if the consumer cannot violate the government
Chapter1: Making Economics Decisions
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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.
Since preferences are monotone, this implies strict convexity.
(b) Find the consumer's Marshallian demand if the consumer cannot violate the government
limit.
(c) Find the consumer's expenditure function.
(d) 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 p₁ 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?
(e) Suppose that, with the limit in place, the price of good 1 increases. If the consumer
chooses ₁ = ₁ both before and after the price change, how does the compensating
variation compare to the equivalent variation of this price change?
£1
(f) Consider a different consumer (i.e. one who may not be maximizing the given utility
function). Suppose that, when facing the same limit of ₁ on good 1, the consumer
chooses ₁ <₁. If this consumer's choices satisfy the weak axiom of revealed preference,
is it possible that she will choose ₁ > ₁ after the restriction is removed if, at the same
time, the price of good 1 increases (and the price of good 2 remains the same)? You
may assume that p₁₁ <w.
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