Problem Set 3 - Solution - Econ 313-1 TA: Tian Liang...

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Econ 313-1 TA: Tian Liang Solutions to Problem Set 3 1. i. Demand for goods x and y as functions of p x , p y and m a. max x;y x 3 y s.t. x > 0 ; y > 0 p x x + p y y 6 m 3 y x = p x p y p x x + p y y = m = ) ( x = 3 m 4 p x y = m 4 p y b. max x;y 3 ln x + ln y s.t. x > 0 ; y > 0 p x x + p y y 6 m 3 y x = p x p y p x x + p y y = m = ) ( x = 3 m 4 p x y = m 4 p y Alternatively, 3 ln x + ln y = ln ( x 3 y ) , so 3 ln x + ln y is a monotonic transfor- mation of x 3 y and thus represents the same preferences as x 3 y . So parts a and b must have the same results. c. max x;y 3 ln x + y s.t. x > 0 ; y > 0 p x x + p y y 6 m Interior solution: 3 x = p x p y p x x + p y y = m = ) ( x = 3 p y p x y = m p y 3 So if m=p y < 3 , the maximization problem has a corner solution where x = m=p x and y = 0 . In summary, the solution to the utility maximization problem is: if m p y > 3 , x = 3 p y p x , y = m p y 3 ; if m p y < 3 , x = m p x , y = 0 : 1
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max x;y 3 x + y s.t. x > 0 ; y > 0 p x x + p y y 6 m If p x =p y > 3 , x = 0 , y = m=p y ; if p x =p y < 3 , x = m=p x , y = 0 ; if p x =p y = 3 , any ( x ;y ) on the budget line p x x + p y y = m is optimal. e. max x;y min f 3 x;y g s.t. x > 0 ; y > 0 p x x + p y y 6 m 3 x = y p x x + p y y = m = ) ( x = m p x +3 p y y = 3 m p x +3 p y ii. Income o/er curve and Engel curve when p x = 2 and p y = 1 a. x = 3 m 8 , y = m 4 Income o/er curve:
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Problem Set 3 - Solution - Econ 313-1 TA: Tian Liang...

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