401-fin-11wa

401-fin-11wa - 1 8th April 2011 Instructions: Answer 7 of...

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1 8th April 2011 Instructions : Answer 7 of the following 9 questions. All questions are of equal weight. Indicate clearly on the f rst page which questions you want marked. 1. Answer both parts. (a) What does it mean to say that a utility function, u ( · ) , represents a preference relation on some choice set X ?P r o v et h a ti f u ( · ) represents preference relation ± , this preference relation must be complete and transitive. (b) Now suppose X = R 2 + and ( x 1 1 ,x 1 2 ) " ( x 2 1 ,x 2 2 ) when x 1 1 >x 2 1 , or x 1 1 = x 2 1 and x 1 2 >x 2 2 . Is this preference relation complete, transitive and continuous? Defend your answers. ANSWER (a) A utility function u : X R represents a preference relation ± on the choice set X if x, y X, x ± y u ( x ) u ( y ) . Completeness: We need to show that x, y X ,e ither x ± y or y ± x or both. Now note that u ( x ) and u ( y ) are real numbers so either u ( x ) u ( y ) in which case x ± y or u ( y ) u ( x ) in which case y ± x ;i f u ( x )= u ( y ) we know both are true. Transitivity: Suppose x,y,z X ,and x ± y and y ± z. We need to show x ± z. Since u represents ± ,u ( x ) u ( y ) and u ( y ) u ( z ) . Since these are real numbers u ( x ) u ( z ) x ± z. (b) This is an example of lexicographic preferences where good 1 is the dominant good. Lexicographic preferences are complete and transitive but not continuous. Completeness: Consider any two distinct points in R 2 + ( x 1 1 ,x 1 2 ) (point a )and ( x 2 1 ,x 2 2 ) (point b ). If x 1 1 >x 2 1 ,a " b. If x 2 1 >x 1 1 ,b " a. If x 1 1 = x 2 1 , since a and b are distinct, it must be that either x 1 2 >x 2 2 in which case a " b or x 2 2 >x 1 2 in which case b " a. Transitivity: Let a,b,c R 2 + where a =( x 1 1 ,x 1 2 ) ,b =( x 2 1 ,x 2 2 ) and c =( x 3 1 ,x 3 2 ) . Suppose a " b and b " c. We want to prove a " c. Now notice that a " b and b " c x 1 1 x 3
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2 x 1 1 = x 2 1 = x 3 1 in which case it must be that x 1 2 >x 2 2 >x 3 2 , and thus x 1 2 >x 3 2 which means a " c. In case (ii) x 1 1 >x 3 1 andthenimmed iate lyweknow a " c. Continuity: Suppose { x n } R 2 + and lim n →∞ x n = x and { y n } R 2 + and lim n →∞ y n = y and x n " y n , n. Then if " were continuous we would be able to deduce that x " y. Here is one counterexample. Let x n =(1 /n, 0) and y n =(0 , 1) . Then x =(0 , 0) , y =(0 , 1) ,x n " y n n but y " x. So lexicographic preferences are complete and transitive but they violate continuity. 2. This question applies what we have done in class this term to think about a car- bon tax. Suppose a typical Canadian household currently buys 200 litres of gasoline per month at $1 per litre and spends $1800 per month buying goods (and services)
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401-fin-11wa - 1 8th April 2011 Instructions: Answer 7 of...

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