SI_562_Assignment_1_AnswerKey

SI_562_Assignment_1_AnswerKey - (y 2 8x − L’ 1(x y =...

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Homework Assignment 1 (10 th edition P67-69: #3, 4, 5, 7, 8) 3. a) 2 5 . 1 12 100 Q Q dQ dTC + - = b) Q Q dQ dATC + - - = - 6 50 2 c) Q dQ dMC 3 12 + - = d) 75 . 0 - = dQ dP e) 5 . 0 6 . 0 X dX dQ = 4. a) Y’ = (8X 4 – 6X 2 )/(16X 4 – 8X 2 + 1) b) Y’ = 40X 4 – 6X 2 c) Y’ = 128X 3 – 80X 5. a) 2 2 2 20 20 ) 5 20 ( ) 25 ( Q Q Q Q Q Q TC TR - + - = + + - - = - = π b) FOC : 0 4 20 0 = - = Q dQ d π , SOC : 0 4 2 2 < - = dQ d π 5 * = Q c) 30 50 100 20 ) 5 ( = - + - = π ; 20 ) 5 ( = P d) FOC and SOC are the same as b): 5 0 -4 , 0 4 20 0 * 2 2 = < = = - = new new new Q dQ d Q dQ d π π . However, the total profit is decreased by $5, the same as the decrease in the fixed cost. That is, new total profit is 25 ) 5 ( = new π .
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7. a) 2 1 5 . 0 5 Q Q TC + = 1 2 5 . 0 10 Q Q TC + = b) 5 . 0 4 . 0 1 6 . 0 5 . 0 9 . 0 ) 6 . 0 )( 5 . 1 ( K L L K L Q - - = = 5 . 0 6 . 0 1 5 . 0 6 . 0 75 . 0 ) 5 . 0 )( 5 . 1 ( - - = = K L K L K Q c) 4 . 0 2 . 0 3 . 2 1 3 . 1 4 . 0 2 . 0 25 . 3 ) 3 . 1 )( 5 . 2 ( B A A B A A P Y P P P Y P Q - - - - = - = 4 . 0 8 . 0 3 . 1 1 2 . 0 4 . 0 3 . 1 5 . 0 ) 2 . 0 )( 5 . 2 ( B A B A A P Y P Y P P Y Q - - - - = = 6 . 0 2 . 0 3 . 1 1 4 . 0 2 . 0 3 . 1 ) 4 . 0 )( 5 . 2 ( - - - - = = B A B A B A P Y P P Y P P Q 8. a) X Y X Y X Y X S 20 20 10 100 200 ) , ( 2 2 + - - + = Y X X S 20 20 200 0 + - = = , and X Y Y S 20 40 100 0 + - = = 15 * , 25 * = = Y X b) 250 , 3 ) 15 , 25 ( = S 9. Lagrangian: L (x, y) = (x
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Unformatted text preview: (y λ 2 8x) − L’ 1 (x, y, ) = 2x – 2 + 8 λ λ L’ 2 (x, y, ) = 2y - 2 y λ λ L’ λ (x, y, ) = 8x – y λ 2 Thus, x = 0, y = 0, = ¼ λ min x,y (x 1) − 2 + y 2 = 1 Substitution: y 2 8x = 0; (x 1) − − 2 + y 2 (y 2 /8 1) − 2 + y2 ……(x = y 2 /8) y 4 /64 + 3y 2 /4 + 1 y’ = y 3 /16 + 3y/2 y’ = 0 y 3 /16 + 3y/2 = 0 y = 0 x = 0 min x,y (x 1) − 2 + y 2 = 1...
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