Throughout this problem set, let X = R2, and suppose a consumer faces a price vector p= (P1, P2) and has an income m, where P₁, P2, m > 0.

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Chapter1: Making Economics Decisions
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Throughout this problem set, let X = R2, and suppose a consumer faces a price
vector p = (P1, P2) and has an income m, where p1, P2, m > 0.
1. [This question gives an example in which the consumer does not have an optimal
solution.]
Suppose (m, P1, P2) = (2, 1, 2) and u takes the form
u(x) =
x1 + x₂
1-7
if ₁ < 2
if ₁ > 2.
Show that:
(a) If r = (2,0), then r is not optimal for the consumer
native affordable bundle yields a strictly higher utility.
(b) If x EX (p, m) is not equal to (2, 0), then à is not optimal for the consumer.
Conclude that the consumer has no optimal consumption bundle in X (p, m).
that is, some alter-
Transcribed Image Text:Throughout this problem set, let X = R2, and suppose a consumer faces a price vector p = (P1, P2) and has an income m, where p1, P2, m > 0. 1. [This question gives an example in which the consumer does not have an optimal solution.] Suppose (m, P1, P2) = (2, 1, 2) and u takes the form u(x) = x1 + x₂ 1-7 if ₁ < 2 if ₁ > 2. Show that: (a) If r = (2,0), then r is not optimal for the consumer native affordable bundle yields a strictly higher utility. (b) If x EX (p, m) is not equal to (2, 0), then à is not optimal for the consumer. Conclude that the consumer has no optimal consumption bundle in X (p, m). that is, some alter-
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