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Unformatted text preview: Math 10260 Final Exam Solutions  Spring 2009 1. Since f ( x ) is a probability density function we have R 3 f ( x ) dx = 1 . 2 = 0 . 8. 2. P (3 < X < 4) = 0 . 2 3. The producer surplus (PS) is equal to the area of the rectangle [0 , 8] [0 , 5] minus the area under the supply curve. That is, PS = 40 R 8 S ( q ) dq . Using the indicated Riemann sum to estimate the integral in this formula we obtain PS = 40 [2 2+(2 . 5) 2+3 2+(3 . 75) 2] = 40 22 . 50 = 17 . 5. 4. Equilibrium happens when demand equals suppl. Thus setting D ( q ) = S ( q ) gives 81 q +3 = q + 3, or ( q + 3) 2 = 81, or q + 3 = 9, or q = 9 3. Choosing the positive solution we obtain that q e = 6. Then, the equilibrium price is p e = S (6) = 6 + 3 = 9. 5. At any time t (during the duration of the mortgage) the rate of change of M ( t ) is equal to rate at which it increases due to interest minus the rate of payments. In symbols, this reads as: dM dt = 0 . 05 M 18 , 000 . It is given that M (0) = 200 , 000. 6. We must compare the PV of option (I) with the amount offered in option (II). We have PV = Z 10 10 , 000 e . 1 t e . 05 t dt = 10 , 000 Z 10 e . 05 t dt = 10 , 000 e . 05 t . 05 10 = 200 , 000( e . 5 1) 129 , 744 . Therefore option (I) is more beneficial. (Note: We could reach the same conclusion by computing FVs.) 7. We have MPK = P K = 18 1 3 K 2 3 L 2 3 , which for K = 1 , 000 and L = 27 , 000 is equal to 6(1 , 000) 2 3 (27 , 000) 2 3 = 3 900 100 = 54....
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 Fall '09
 HIMONAS
 Probability

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