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562_f08_final_solution - SI 562 Final Exam Solution Fall...

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SI 562 Final Exam Solution: Fall 2008 1. Mary’s utility function is U(b,c)=b+100c-c 2 , where b is the number of silver bells in her garden and c is the number of cockle shells. She has 500 square feet in her garden to allocate between silver bells and cockle shells. Silver bells take up 1 square foot and cockle shells each take up 4 square feet. She gets both kinds of seeds for free. a. To maximize her utility, given the size of her garden, how many silver bells and cockle shells should Mary plant? b. If she suddenly acquires an extra 100 square feet for her garden, how much should she increase her planting of silver bells and cockle shells? Use the Lagrangian to solve this part. Solution: a. To maximize U(b,c)=b+100c-c 2 subject to the constraint: b+4c<=500 . Plug b=500-4c into U(b,c) : U(c)=500 – 4c +100c - c 2 = 500 + 96c – c 2 Take the derivative respect to c and set it to 0 (maximization): dU dc = 96 - 2 c = 0 2 c = 96 Þ c = 48 Check the SOC, 0 2 2 2 < - = δχ Υ δ . Plug 48 into b=500-4c : b = 308 b. Now the constraint changes to: b+4c<=600. Objective function: U(b,c)=b+100c-c 2 Unit cost of b : 1 Unit cost of c : 4 Cost constraint: 600 L Q = U - l ( Cost b b + Cost c c - Const ) L Q = b +100 c - c 2 - l ( b + 4 c - 600) L Q = b +100 c - c 2 - l b - 4 l c + 600 l d ( L Q )/ db =1- l = 0 (1) d ( L Q )/ dc =100 - 2 c - 4 l = 0 (2) d ( L Q )/ dl = 600 - b - 4 c = 0 (3) From (1), we get λ = 1 Plug λ = 1 into (2): 2 c = 96 Þ c = 48 Plug c = 48 into (3): b = 408 So Mary should plant 100 more silver bells while keep the same amount of cockle shells.
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SI 562 Final Exam Solution: Fall 2008 2. (a) 0 ] 50 5 10 [ & 0 ] 50 5 10 [ 5 5 100 95 100 25 50 5 10 . . 5 5 100 95 100 25 2 1 2 1 2 1 2 2 2 1 2 1 , 2 1 2 1 2 2 2 1 2 1 , 2 1 2 1 = - + - + - - - - + + - = = + - - - + + - = Π Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Μαξ Θ Θ τ σ Θ Θ Θ Θ Θ Θ Μαξ Θ Θ Θ Θ λ λ λ L (b) You can use the Lagrangian or substitution methods. 455 . 9 (4) into Plug 273 . 0 220 60 0 90 30 150 190 (5) (4) (5) 0 90 15 190 10 20 190 5 200 100 (2) (1) (4) 2 10 (3) (3) 50 5 10 0 (2) 5 5 10 95 0 (1) 10 5 200 100 0 : 2 1 1 1 2 1 1 2 2 1 1 2 2 1 1 2 2 1 = = = = + + - + = + - - = - - + - = + - - = = - - - = = - - - = = Θ Θ Θ Θ Θ - Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ Θ ΦΟΧ 1 2 1 Q L Q L Q L l l 3. Between 1999 and 2000, the quantity of automobiles produced and sold declined by 20 percent. During this period the real price of cars increased by 5 percent, real
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