finalans - CORNELL UNIVERSITY FINAL EXAM Wednesday,...

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CORNELL UNIVERSITY FINAL EXAM Wednesday, December 16, 2009 INTERMEDIATE MICROECONOMIC THEORY ECON 3130 Professor Majumdar Fall 2009 You have 150 minutes to complete this exam. There are 135 points. Permitted Materials: Non-graphing calculators only. WARNING The exam is divided into two parts, Part A and Part B. You must answer failure to do so will result in a 10-point penalty. On the cover of each exam booklet, please state whether it includes Make sure your name is on all exam booklets used. When time is called, please put your answers to Part A in the box marked ±Part A², and put your answers to Part B in the box marked ±Part B². No exam booklets will be accepted after we leave the room. PLEASE DO NOT OPEN THIS EXAM UNTIL INSTRUCTED TO DO SO Page 1 of 10 ECON 3130
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Question 1 Answer "yes" or "no" with a brief justi±cation of your position ( a ) [ 5 points ] No, it is not true in general. As an example, take any smooth utility function over 2 goods (say, the Cobb-Douglas utility function u ( x;y ) = x 1 2 y 1 2 ). Then, consider any bundle on one of the axes but outside of the budget set (for example, 2 m p x ; 0 ± ). It then follows that u ( x ;y ) = ( x y ) 1 2 > u 2 m p x ; 0 ± = 0 . ( b ) [ 5 points ] No, both goods x and y cannot be both Gi/en goods. If both are Gi/en then by de±nition both must be inferior goods, something we have already proven cannot hold: Assume both goods x and y are inferior. De±ne ( x ;y ) to be the optimal bundle of goods chosen given the budget set B ( p x ;p y ;m ) and consider some variation of income. Since both goods x and y are inferior, then by de±nition of an inferior good we have @x ( p x ;p y ;m ) @m < 0 and @y ( p x ;p y ;m ) @m < 0 . By the assumption that "more is preferred to less" of both goods, we know that the budget constraint has to bind at the optimum. In other words, p x x ( p x ;p y ;m ) + p y y ( p x ;p y ;m ) = m: Di/erentiating implicitly with respect to m , we get p x @x ( p x ;p y ;m ) @m + p y @y ( p x ;p y ;m ) @m = 1 . However, since p x and p y are de±ned to be positive, then @x ( p x ;p y ;m ) @m < 0 and @y ( p x ;p y ;m ) @m < 0 imply that p x @x ( p x ;p y ;m ) @m + p y @y ( p x ;p y ;m ) @m < 0 6 = 1 : ( c ) [ 5 points ] No. Consumer 2 will choose the same commodity bundle ( x ;y ) : v ( x;y ) = 2 u ( x;y ) + 3 is a linear transformation of u ( x;y ) , and therefore it is a monotonic increasing transformation of u ( x;y
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finalans - CORNELL UNIVERSITY FINAL EXAM Wednesday,...

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