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WeekofFeb02_solns - W3211 Spring 2009 Professor Vogel...

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W3211 Spring 2009 Professor Vogel Recitation for the Week of February 02, 2009 . 1. (a) MRS = MU x =MU y = 1 =x 1 = 1 =x , It is diminishing with x but is independent of y: (b) max x;y;° L = y + ln x + ° (12 ° 2 x ° 3 y ) @ L @x = 1 x ° 2 ° = 0 @ L @y = 1 ° 3 ° = 0 @ L = 12 ° 2 x ° 3 y = 0 By °rst two FOCs, ° = 1 = 2 x = 1 = 3 ! x ° = 1 : 5 Substituting x ° = 1 : 5 into the last FOC, y ° = 3 2. In order to get elasticities we need demand functions of x and y max x;y;° L = y + ln x + ° ( I ° p x x ° p y y ) @ L @x = 1 x ° p x ° = 0 @ L @y = 1 ° p y ° = 0 @ L = I ° p x x ° p y y = 0 By solving this maximization problem, we have y ° = I p y ° 1 and x ° = p x p y (a) e y;p y = @y @p y p y y = ° I p 2 y p y y = ° I yp y e y;I = @y @I I y = 1 p y I y = I yp y e y;p x = @y @p x p x y = 0 (b) e x;p x = @x @p x p x x = ° p y xp x e x;I = @x @I I x = 0 e x;p y = @y @p x p x y = p y xp x (c) The optimal consumption of x , x ° = p x p y , is independent of income I: Ux p x = 1 p x x ; U y p y = 1 p y
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2 3. By MRS = 0 : 4 x ° 0 : 6 y 0 : 6 0 : 6 x 0 : 4 y ° 0 : 4 = 1 2 = p x p y and his budget constraint 10 = x + 2 y , the initial optimal ( x ° ; y ° ) is (4 : 3) . After the price change of x , p x = 2
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