400ps2-11-soln - ECON 400 Winter 2011 SOLUTIONS FOR PROBLEM...

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E C O N 4 0 0 H a r t m a n Winter 2011 SOLUTIONS FOR PROBLEM SET II 1(a) Since 232 12 1 2 () tx tx t x x , the function is homogeneous of degree 3. Euler's theorem holds because 2 f x and 21 2 2 f xx so that 222 11 2 2 233 x fx f x x x x x x f  . 1(b) Since 22 2 2 1 ()() ( ) tx tx tx t x x x  , the function is homogeneous of degree 2. Euler's theorem holds because f x and 2 2 f  so that 2 2 1 2 2 (2 ) 2 ( ) 2 x f x x x x f . 1(c) Since 1 1 2 2 2 2 2 2 1 2 1 2 () 2 ( 2) 2 tx tx t x x x x t tx tx t x x x x       , the function is homogeneous of degree 1 . Euler's theorem holds because 121 1 2 1 2 ( ) ) x x x f and 122 1 2 2 2 24 ( ) ) x x x f  so that 1 1 2 1 2 1 2 2 2 2222 1 2 1 2 1 2 2 2 ( ) ( ) ) ( ) (224 ) ( ) ( 1 ) ) ( ) (1 ) . ) ) 2 x xx x x xf x x xxxx x x x x x x x x     1(d) Since 2 2 2 0 1 1 2 2 2 2 1 1 1 ( ) tx t x x x t tx tx tx t x x x x x x x x x , the function is homogeneous of degree 0.
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