复旦大学：《高级微观经济学—均衡理论》英文版 Problem set 1 Prof gene Chang

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). Th

Problem 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)

weakely consumption sets are convex. Draw typical indifference curves in a diagram, with xI on the vertical axis and x on the horizontal axis. Then answer 1. Is the preference monotonic? Is it of locally non-satiation? 2. When we derive a demand function for xi from this preference, is the demand curve continuous?

weakely consumption sets are convex. Draw typical indifference curves in a diagram, with x1 on the vertical axis and x2 on the horizontal axis. Then answer: 1. Is the preference monotonic? Is it of locally non-satiation? 2. When we derive a demand function for x1 from this preference, is the demand curve continuous?