a2Problem Set 2 Solutions - Econ 201A Fall 2010 Problem Set...

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Econ 201A Fall 2010 Problem Set 2 Suggested Solutions 1. Prove that the lexicographic preference (defined in Example 3.10 in the course notes) is complete, transitive, and antisymmetric (if x % y and y % x , then x = y ). Proof. Completeness Let x, y R 2 be given. If x 1 > y 1 , then x % y . Similarly, if y 1 > x 1 , then y % x . Suppose that x 1 = y 1 . In this case, if x 2 y 2 , then x % y .
2. Prove Proposition 4.12 in the course notes: If u represents % , then: (a) % is convex if and only if u is quasiconcave. (b) % is strictly convex if and only if u is strictly quasiconcave. Proof. Suppose u represents % . Then u ( x ) u ( y ) ⇐⇒ x % y ( x, y X ) , (1) u ( x ) > u ( y ) ⇐⇒ x y ( x, y X ) . (2)
(b) Here we apply both (1) and (2). (Note that the conditions x, y X and α (0 , 1) have been omitted to save space.)
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Econ 201A Fall 2010 Problem Set 2 Suggested Solutions 3. Suppose the consumer’s utility function is Leontief: U ( x 1 , x 2 ) = min { x 1 , x 2 } Prove that x * ( p, w ) is a singleton.
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