Problem set 1 Prof gene Chang Due on dec.5 th in the class.,orTA' s office(经院220室) Late turn- in may result in a penalty in the grade 1. The consumption set is X=x: XER). Suppose the util ity function representing the preference =-1+2x 2<x<∞ as illustrated in the diagram(a). Is the preference continuous? If not continuous, illustrate the set that violates the definition for continuity (a) 2. Suppose the preference is represented by the utility function as illustrated in the diagram(b),is the preference continuous? And if it is not continuous, illustrate the set that violates the definition for continuity 3. The consumption set is X=x: XE R+. Suppose the utility function is U(x)=U(x1,x2)=x1+x2, except U(2,2)=3 Suppose the consumer's preference is represented by this utility function. Prove this preference is complete, reflexive, transitive, but not continuous and monotonic.(Notice this time you are working with the preference, not with the utility function. You should use the definition for the continuity for the preference given in the class to give a proof. 4. Suppose the consumption set of a consumer is X=x: xE R), and xi is a normal good and x is a bad(bad is something like garbage ---the more a consumer has, the worse off he is). ThProblem Set 1 Prof. Gene Chang Due on Dec. 5th in the class, or TA’s office(经院 220 室). Late turn-in may result in a penalty in the grade 1. The consumption set is X R { : } =  + x x . Suppose the utility function representing the preference is 2 0 2 ( ) 1 2 2 x x U x x  =     = − +    x as illustrated in the diagram (a). Is the preference continuous? If not continuous, illustrate the set that violates the definition for continuity. 2. Suppose the preference is represented by the utility function as illustrated in the diagram (b), is the preference continuous? And if it is not continuous, illustrate the set that violates the definition for continuity. 3. The consumption set is { : } 2 X = x x R+ . Suppose the utility function is 1 2 1 2 U(x) =U(x , x ) = x + x , except U(2,2) = 3 Suppose the consumer’s preference is represented by this utility function. Prove this preference is complete, reflexive, transitive, but not continuous and monotonic. (Notice this time you are working with the preference, not with the utility function. You should use the definition for the continuity for the preference given in the class to give a proof. ) 4. Suppose the consumption set of a consumer is { : } 2 X = x x R+ , and x1 is a normal good and x2 is a bad (bad is something like garbage --- the more a consumer has, the worse off he is). The X U(x) (b) X U(x) (a)